> we can go very far with very little funding, so why sell shares when we can get a 50k grant in exchange for some (admittedly tedious) paperwork?
To be entirely frank, it depends on the field, but I imagine it's easier in many ways to get VC funding than a grant for many projects. Grants are a pain to apply for (there's a reason that full-time grant-writers exist) and the feedback loop is very slow. Compare that to the VC world, where you can quite easily raise a small amount of seed/pre-seed capital quickly, and even when you're rejected, you're likely to hear back quickly[0].
This will be a challenge for YC, because it inherently creates an adverse selection problem. The companies that YC (and all venture funds) want to attract the most are the ones who could win a grant if they applied, not the ones who are seeking venture funding simply because they can't actually survive the grant process[1]. The grant application process is terrible for a lot of reasons, but it does have the side-effect of filtering out a lot of people who simply aren't that serious or invested in their work. For YC, it means a lot more companies to sift through to identify the gems.
There are certainly some billion-dollar companies that could not survive the grant process (or simply would prefer not to go through it), but if we're looking strictly at the ratios, it's going to be a lot more challenging for any venture fund to find these 'needles', because the 'haystack' is simply much larger (relatively speaking).
[0] If you never hear back from a VC, that means you're probably either rejected or being punted on ("not now, but maybe later"). But that's a good thing - at least you know! If you don't hear back from a grant application after 2 months, that doesn't actually mean anything other than your application may not have been reviewed yet.
[1] Possible reasons for failing to win grants: project/business model is simply not valuable, project is not well fleshed-out, company does not have the runway to survive the drawn-out application cycle, etc. All of these apply to the venture world as well, but the cycle there is much faster.