Of course it can, that's what most public transport companies are. They tend to be government subsidized. There's very little competition because there is not

Of course it can, that's what most public transport companies are. They tend to be government subsidized. There's very little competition because there is not much to compete for, except in longer distance rail and bus. You also need permission from the city to use public stops, consult routes and times, which is a barrier of entry. Not an insurmountable one and nobody prohibits you from running private stops or garages. You can even negotiate the use of bus lanes or tram lines. The other barrier is much higher capital investment. Both bus drivers, busses and stops are not cheap, while the margin on a ticket is thin. (Compared to taxi fleet.) But it can be done even encouraging competition - see South Korea for an example of where it worked.