Such hard evidence probably does not currently exist. Recently, however, a random "funding lottery" scheme has been implemented in New Zealand, and is being studied. However, the sample size is small, and it may take 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
- The Health Research Council (HRC) of New Zealand's “Explorer Grants”.
- New Zealand’s Science for Technology Innovation's (SfTI) “Seed Projects”.
- 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.
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 <...>