Exactly. Everyone I know here with a driver owns the car themselves and hires the driver separately. I don't know driver salaries since I don't have one but I'm guessing they are similar to nanny salaries since they are usually full-time 8am-8pm, 6-days a week kind of jobs. The only people without their own car tend to be foreign executives who are only going to be here on a 2-3 year contract.
Driver salaries would likely have been much higher because the position wouldn't also cover room & board. Also the gender differences in pay would influence things as well. Last is that household cleaning etc. would have been considered lower skilled labor.