Are you from the UK? I feel you may have a higher standard of taxi drivers that is far above expectations that you would expect stateside.
If I wasn't comfortable letting my daughter ride an uber, I sure as hell wouldn't be comfortable letting my daughter ride a taxi. Thus far, Uber has consistently provided a far better and comfortable ride than a standard taxi. If regulation hasn't prevented taxi drivers that have, in my experience talked on the phone while driving, lied to me and said they don't accept credit cards, or are just plain hard to flag down. From my, single POV, Uber drivers are that much better and the belief that regulation has 1.) created an acceptable minimum taxi service (it hasn't) 2.) will ensure Uber has a high quality (it already does) is one I can't fully swallow.
Uber already does a great job of vetting their drivers (I've heard drivers who receive even a few bad ratings are sacked) and honestly if the service was so dangerous it wouldn't be such a sought after alternatives to Taxis.