AV content is pretty important to humanity. I want my government to tax me and spend the money on a public video repository.
When some garbage for-profit outfit like Alphabet is in charge of the world's video, I expect ads and garbage content.
The issue of private- vs public-run projects is complicated by culture. There's a vicious cycle / virtuous circle that happens in a given society. In a place where government projects have a bad reputation (of being wasteful and substandard), it makes it difficult for government projects to succeed.
That's a long way of replying to you that you're probably correct about cost-overruns in most nations - but I don't believe it has to be that way. You can probably predict the nations I'd trot out that do a better job.
At any rate, where Youtube is concerned, I'd sooner government be in charge, regardless of cost. I don't see Youtube ever making the public good its primary focus - not if its front page, recommendations, or comments section are any indication.