For many pricing problems, auctions are the answer. Doing a roadshow and tickling investor's interest is a separate problem, but many startups already have a size where they are well-known to a big enough share of investors.
There’s a lot of auction theory that comes into play that an individual investor is unlikely to know and I would imagine a startup may not fully comprehend either. I would worry that this is still not a level playing field.
This will force investment banks to have an underwriting fee no greater than the expected loss due to the company not understanding "auction theory." Also, any investor that doesn't think they understand auctions can simply wait a day and receive the exact same opportunity that an investment bank would give.