The hosting costs would be astronomical for a startup to offer free HD video hosting in the modern era. Google had deep pockets to buy them and pay for servers for all of the years YouTube was not profitable. A viral video like Gangnam style which currently has 3.6B views probably cost in the millions to serve. Luckily Google has the ad muscle to recoup that, but not many others could. If someone were applying to YC or approaching VCs with a YouTube competitor, the big questions that I'm sure would come up are, "what have you figured out that YouTube hasn't that will make users flock to you" and the obvious followup if you have a clever answer to that: "why couldn't Google beat you by just doing that first or better?"

EDIT: thinking about it further, the company most suited to offer it would be Amazon, which obviously has the storage space and hosting infrastructure, but not the advertising network to make revenue on it. It would be interesting if they could tie into Prime Video to allow users to monetize their content via a revenue split with Amazon.