Wall-street/VC's is focused on convincing people and well funded organizations to part with their money. Thus they need to be able to talk up their choices. Even if it's just "we only employ people from MIT and Harvard so can trust us." But, everyone does that so they really want to be able to say: "I had the foresight to invest in Google so you should trust my judgement."
That said, they could also just randomly choose without telling anyone and justify actions after the fact. But, as you might guess nobody who does that is going to say they did it that way.
The basic problem is if your going to randomly pick then why should I pay you anything?
PS: One fund did chose to fund every YC company which is close to a random choice.
Just to throw something back to your question: Because you showed that randomized investing is a net win, and that your randomization methods are robust. Of course I'm still very curious if this is the case about randomized investing being a net win (though my instinct is that investing in startups is no different than investing in the stock market, and therefore at least as good as randomly investing, because no one has a crystal ball). But if you can just literally show people the numbers... "Hey, put $50k in the pot, X years later, you get $85k out." Or whatever, of course. I just think it would be worthwhile -- seeing that the government already does invest in the stock market -- for the same to feed startups.
As someone else said, you want some vetting to avoid obvious scams. But, in terms of picking the 'best' 5 out of 10 when it's going to take a lot of effort.
Alternatively, a common approach when getting a lot of applications is to simply randomly look at half and ignore the other half unless you can't find enough good options.
So, in the real world you would be able to talk up various points. However, I don't think you can convince a politician to use your random methods unless he can sneak a few choices in.
That said, they could also just randomly choose without telling anyone and justify actions after the fact. But, as you might guess nobody who does that is going to say they did it that way.
The basic problem is if your going to randomly pick then why should I pay you anything?
PS: One fund did chose to fund every YC company which is close to a random choice.