There are two differences between this and the Monsey bus, I think (although I’ll admit I’m not positive about either).
First, in order to pick people up at stops on the street (as opposed to travel from pickup at point a to drop off at point b) the company needs the franchise from the city to allow them to what is in effect compete with the MTA. They could reorganize as a private bus company, but I think would be unable to make stops along the route if they did.
Second, the Monsey bus is an interstate route and is licensed by the USDOT. Not sure if that matters in terms of what is considered discrimination, but it does affect what would or would not be considered a public vs. private route.