Lyft, which made the above qoute and Uber, obviously could continue operating if breifly pausing to fingerprint their drivers. However, the first answer to your question is that they won't. The premise of ridesharing is about scalable technology that allows people to work together on a platform. It would be an expensive precedent for these companies to have to pay for this cost. This is regardless of how you the consumer feel. There is also a very non-trivial impediment to get drivers to do this, especially if they are barred for something like a felony, even non-violent or not related to a vehicle. The difference is who controls the process, which wasn't stated in the article. I suspect a 3rd prty (e.g. Austin) would want drivers to come into a public place (police staton/town hall) and get fingerprinted. I do not believe that could simply thumbprint a phone.
The second answer is privacy & convenience. A driver is unlikely to want to get fingerprinted and it probably makes little difference. Why should some agency get potentially millions of fingerprints (if this become national) when it would serve no purpose. Maybe they can serve warrants, ect. Again, idk if this is good or bad from a public standpoint but in practice Uber/Lyft already have a car, registration, license, picture, social media and very quickly a review system. So to the extent someone wanted to cheat, it would just be another small ipediment.