As an auction theorist who's biased towards putting things in auctions and as an operator who's been through the venture investment process several times, I'm biased in wanting a more efficient capital formation process with less adverse selection (better outcomes) for both sides. So, I'm doubly biased, but I appreciate why this hasn't been done and that it could be done.
Classical auctions aren't a good fit because they can't handle non-price factors. However, a combinatorial clock auction could handle non-price factors while also improving price discovery. But as Milgrom says (paraphrased), the best auction is the one people use, so the more realistic outcome is what OP describes: running your own (implicit) auction.
Classical auctions aren't a good fit because they can't handle non-price factors. However, a combinatorial clock auction could handle non-price factors while also improving price discovery. But as Milgrom says (paraphrased), the best auction is the one people use, so the more realistic outcome is what OP describes: running your own (implicit) auction.