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No such study exists.

You have to realize that the current model of funding research through grants is rather recent. Fifty years ago, research was funded with recurring credits: a lab/researcher got a fixed amount of money each year to do research as they saw fit. Only applied researchers got industrial grants to develop precise new applications, but this wasn't the core of their funding.

The current system arose due to the combination of two factors:

• The growing hegemony of the neo-liberal ideology in the 80s-90s and the commodification of society as a whole. According to this ideology, research as a whole is supposedly more efficient (for an ill-defined notion of "efficient") if researchers compete with one another on the global market of research funding. They will "naturally" tend to research what works in order to keep their funding.


• Politicians' will to govern research. Research, and especially fundamental research, is a notoriously difficult thing to govern: you never know where an idea might lead, or what new ideas will come up along the way. You don't know what will work and what won't. You don't know what will be applicable and what will remain a nice theoretical idea. Otherwise it wouldn't be research.
  
  But rulers want their word in this: they want to tell their voters that their policies improved lives in a very direct way as a result of such and such decision they took. This is why words like "governance" and "steering" are becoming more and more prevalent in modern academia: administrators are not in the business of just helping researchers carry out their research, they are in the business of ensuring that researchers are researching what they are supposed to. And how do you force someone to do what you want, short of physical violence? Through economical violence.

Of course, this yields absurd situations, such as the fabled year when the French National Research Agency spent more on administration than on research. And such thinking paves the way for short-term research with zero long term goals, and especially not goals reachable in more than five years (the usual length of "long-term" grants). But these decisions are not driven by rational thought; they are driven by ideology.

I don't know about such studies, but I have served on ~20 panels to review proposals. While I'm entirely willing to believe that multiple panels will not agree on the relative ordering of proposals, I am quite convinced that they will in general agree on which proposals are "good" and which are "not good".

In any given round, a panel (at the National Science Foundation) will review on the order of 20 proposals. What may not be obvious to outsiders is that of these 10 are pretty obviously not fundable. 3-5 are eventually rated as excellent and the remainder as "pretty good". I am pretty convinced that if you ran multiple panels on these 20 proposals, that this classification into three groups will be more or less stable. What I am also convinced of is that panels will produce different rankings within each group.

The consequence is that it is quite possible that different panels will results in a different set of proposals being funded, given that only ~4 proposals out of the 20 will be funded. But overall, the result will still be far from random. The top 3-5 are still likely to be funded, the bottom 10 will definitely not be funded, and it's a toss-up in between.

There is another aspect to the question at hand. A random assignment of monies to projects can be expected to fail because the system in place, whatever it is, induces certain behaviors. One wants to design a system so that positive behavior is encouraged and negative behavior discouraged.

In a system of peer review those seeking funds are induced to give their analysis of why their proposed project should be funded and why it is highly likely to be successful. They know that their words will have consequences and that the review will be carried out by knowledgeable people who have assumed a certain responsibility.

But if the system were random, and known to be random, the behavior induced would be quite different. There would be no real need to spend the time and effort to examine and explain the background and methodology to be used and to guarantee that it meets scientific rigor. The consequence would be that many more proposals would be made and most of them would be junk.

With that background, note that the proposals that are actually made to funding agencies are pretty much all guaranteed to be at least adequate. So the peer review boards are choosing from among the best proposals that might be expected, rather from the trash that might be thrown out by charlatans.

This brings me to a thought experiment that I've used in other contexts. Suppose, in a system that repeats, you rank a set of things on some linear scale from best to worst, and you discard the worst. The worst is replaced. This is sometimes what happens in companies that rank employees on some numeric scale and fire the worst performer.

But, if you started out with the, say, ten best performers in the world, your system will only worsen your overall system, since everyone else will be worse than the ten you start with.

Thus the question asked by the OP doesn't occur in a vacuum. Nor, I suggest, can the various proposals be ranked in theory on a best to worst scale. There are too many variables and something better on one scale isn't as good on another. So, while the system has flaws, it is in the nature of the world that some flaws will exist. But a system that induces and rewards good behavior is, in principle, about as good as we can hope for.

The various judgments made by the review boards are, in the larger sense of things, relatively minor.

Short answer:

Such hard evidence probably does not currently exist, but finally a random "funding lottery" scheme has been implemented in New Zealand and is being studied. However, the sample size is small, and it may until at least 2026 before real data is obtained.

Why there is a lack of evidence:

In the RAND report Alternatives to Peer Review in Research Project Funding: 2013 Update it was noted that "the idea of randomly allocating research funding has been developed theoretically and is used by some institutions for small grants". While it's clear that the award of said small grants avoids some of the biases inherent in peer review-based funding and much of the administrative burden, the fact that the grants are small (e.g. some are appropriately sized for travel funding) means that the one cannot really infer anything about large-scale random funding schemes.

Things have evolved somewhat since. A 2018 paper titled "Policy Considerations for Random Allocation of Research Funds" by Shahar Avin notes that at least three major funding bodies have started allocating some funds using a degree of randomization. This includes

1. The Health Research Council (HRC) of New Zealand's “Explorer Grants”.


2. New Zealand’s Science for Technology Innovation's (SfTI) “Seed Projects”.


3. Volkswagen Foundation’s “Experiment!” grants.

HRC describes the Explorer Grants as follows:

Explorer grants support transformative research ideas that have a good chance of making a revolutionary change to how we manage New Zealanders' health. They are available in any health research discipline and are worth $150,000 for a term of up to 24 months.

...
Applications for explorer grants are assessed by subpanels within the HRC's Explorer Grant Assessing Committee to see if they meet the criteria of being both transformative and viable. Unlike with any of our other grants, the assessment process for explorer grant applications is anonymous and all applications that meet the criteria are equally eligible to receive funding. A random number generator prioritises these applications.

The HRC does not use this scheme for other grants, but believes

that random funding is a fair and transparent way to choose between equally qualified applicants, and it's particularly suited to Explorer Grants where it may not be appropriate to rank or score high-risk applications with less predictable outcomes.

The SfTI Seed project proposals similarly undergo an initial assessment, and then a some of them form a special pool, from which proposals are randomly drawn. Proposals for the Volkswagen Foundation's Experiment! are screened by an interdisciplinary jury that funds some proposals, rejects some, and leaves some for a funding lottery.

As such, HRC's Explorer Grants are the most interesting from this aspect, as all proposals meeting the minimum requirements go into the lottery. This is why they are currently being studied:

A funding lottery creates a perfect randomized trial because we have equally worthy researchers who are funded at random. We can then track their careers from the point of randomization and compare them in terms of metrics such as publications, citations, and other funding, as well as perhaps more-complex outcomes, such as innovation. We are currently following researchers who applied for funding with the New Zealand Health Research Council and were randomly allocated funding (3); however, the sample size is small, and it may be at least a decade before we have accumulated enough data to show meaningful differences.

Summary:

In conclusion, there does not seem to be any hard evidence that the grant peer review system performs better than a process with random selection for proposals meeting minimum standards. (Like the other answer writers, I'm fairly sure a random system with no quality checks would lead to opportunistic low-standard applications...) The reason is that such randomized funding has basically not been attempted. The same letter cited above hints at the real reason why this hasn't been studied more, despite lacking evidence for the peer review-based system:

We have spoken with Australian funding agencies about using a lottery, and the reaction was strongly negative, with one staff member saying, “It would make it look like we don’t know what we’re doing.” A key concern is that politicians and the public would react negatively <...>

This will only be a supplement to the answer here of user106886. That answer discusses an interpretation of the actions of political actors. However, not all funding is from the government, so I'll write here about another option: private funding.

Back during the Renaissance, artists and scientists were funded by patrons, often kings and princes, who controlled vast sums. There is still a vestige of patronage now in some circles. Web comics can be funded now, for example, by Patreon, thorough which people (like myself) pledge a few dollars a month to support the creators of the comics. We, the patrons, have no real influence on what gets created as each is a small part of a larger whole.

But most private funding is from profit making companies, usually publicly traded companies who must justify expenditures. The rules of the game are that those deciding how to spend money have a fiduciary responsibility to shareholders that monies are properly spent in furtherance of the goals of the company. Therefore, money can't just be given out randomly, but requires some assurance that it will be well spent. This results in a pre-review of some kind, though not exactly the same as that required for government funding.

There is a review and it is similar, at least, to peer review. It could even be peer review, as most companies wanting to fund research in X normally also have research departments skilled already in X. If a company gets 10 proposals and can only fund 3 of them, it needs to do a cost-benefit analysis before it can award funds, even if the analysis is flawed. It has to appear to be valid on its face.

Bill Gates and Elon Musk, could, in principle use their own money to conduct a random experiment, but Microsoft and Tesla cannot. If the company gives you money they need some expectation that it will be properly spent and not wasted.