Renting a car would be a great deal cheaper than hiring a private driver.
I don't think you'll find a regular tour operating from Ronda to the white villages. There are one-day tours from Seville that go to Ronda and one or two other places. There may also be such tours from Malaga. I didn't consider those viable options, because I didn't want to rush through Ronda in an hour or two. If you want to have a decent amount of time in Ronda, you kind of need to do what you are doing--staying in Ronda itself.
I know one of the popular small villages is accessible as a day-trip from Ronda by public bus [edit: in April; cannot vouch for the off-season]. I don't remember whether that is Grazalema or Zahara de la Sierra; I'm out of town at the moment and don't have access to my trip notes.
If you want to use public transportation, I believe Arcos de la Frontera is most easily visited from Jerez; it would be farther from Seville or Cadiz, probably beyond reasonable day-tripping distance. If starting in Ronda, I think you'd first have to travel to one of those places, making it a horribly long (or perhaps impossible) day-trip.