Canada

In general yes, however just like the feds are starting to work directly with municipalities on housing because provinces are dropping the ball, it's conceivable that the feds could contribute to postsecondary institutions too. AFAIK the federal government contributes to UofT's funding for example.

Public universities are nonprofits. The money they "make" is to cover their budgets which do a lot. Irrespective of how much staff they have, the source of funding inevitably changes the focus of the school. For example focusing on international students and the programs that are popular among them because their tuition isn't regulated and can be several times higher than domestic. This could come at the expense of funding other programs that aren't as popular while still import for society. Let me repeat, public universities are coming up with alternative sources of income to replace ever reducing public funding. If the trend continues, eventually the affordability will go away too. It already has for certain domestic programs. The end result is sky high tuitions and student debt levels like you have in the US. And make no mistake, we won't ever need less postsecondary education and research in the coming decades in our society if we want to keep a decent standard of living.

Universities make money hand over fist

Proof please.