While what you say is true in principle, in practice it would quickly translate into deteriorating service.
Public transit is and should of course always be mostly tax funded, but a small fare can add money in the system, it adds demands from users and a sense of responsibility for the suppliers.
The fare is always much cheaper than operating a car and its easy to have discount programs for those that need it.
Enforcement can be as easy as random checks, all the fancy infrastructure is not really needed although for underground systems its good to have a little barrier for entry.
I can afford a car just fine but I don’t want to spend my money on it just to get to work.
The real measure of the affluence and quality of life in a city is how large a portion of the wealthy take transit. The higher the better.