As far as the ministry's lack of participation in the school... it seems to be that up until Order of the Phoenix, the minister has allowed Dumbledore to make those decisions, but when he thinks enough is enough, he has the power to do all kinds of stuff.
When Vernon says he won't pay for Harry to go to Hogwarts... well, Vernon had no clue what Hogwarts was, really, or how it is run, except that it was a magical school. So that's not really a good backing point.
About Harry saving his money, or Hagrid mentioning that James and Lily wouldn't have left him without some provision to be able to go to school, this is perhaps the strongest evidence, in my opinion, that there is a tuition. But it does say that Harry doesn't want to have to ask for money for spellbooks, with no mention of tuition. Plus, we never hear of Harry going to his vault and paying his tuition. Who knows, maybe it's a direct deposit or automatic transfer, but I somehow doubt it.
What we're having trouble with here is that we are trying to compare a fantastical place, an imaginary place, to something we know in real life. In some cases, that works out great. In this case, I don't think it does. Hogwarts is different than any school ever, it doesn't fall between the lines of "public" and "private", because it's not real and even in the story, it existed before modern rules and standards were made regarding schools.