When public transportation was first introduced in most places, it was run by private companies for profit. This changed mostly because it wasn't profitable to compete with cars when those became popular.
Of course there still are private companies running public transport: long distance buses and trains in many places, and commercial aviation is really also a form of public transportation.
So there is nothing novel about buses being run by private companies for profit.
For me it's the marketing that makes me roll my eyes. Shuttle instead of bus when in the United States. (Curiously, in other countries it's called bus by Uber.)
The only time I hear shuttle used is for a thing that transports between two locations specifically. A "shuttle" from the airport to a hotel or whatever, for example. This seems to match the definition of shuttle also, so I think it's correct. It has nothing to do with marketing, rather actually using the proper term.