Suekay, That is really valuable info. I am not surprised about the lack of taxis since there never has been a lot of good taxi service in the south of France. But, the full bus is a real problem. The only solution would be to pick-up the bus at the Nice bus station, which is where it starts its run. The downside is that the bus station is not very convenient to the Nice train station although it is walkable.
Villefranche is probably our least favorite port in that part of France for several reasons (some of which you have discovered). Tender operations can sometimes be a real hassle (it can take hours for independent passengers to get off the ship). Train service can often involve a 30 - 45 min wait. Bus service on Sundays is also not great. There are no rental car companies close to the port although there have been times when some would deliver to the port (but never on Sundays). We go to great pains to make sure we always have a rental car when we are in the south of France on Sundays, but that does not help those who do not want to drive.