It's worth noting that in addition to competition being somewhat local, Uber and Lyft don't just compete with taxis, but also with public transit and personal vehicles. When competing with public transit, cities should regulate, as there are many public goods arising from quality public transit. The story for personal vehicles however is different, which have most of the same negative aspects of ride-hails, plus some additional negative externalities (parking is a big one). Thus the first step in regulation should be one that targets both personal vehicles and ride-hails.
I support a congestion charge. Even if your preference is strongly toward using a personal vehicle, you should too. The nudge toward users who have a very weak preference toward a personal vehicle will free up road space, creating less traffic for you. Even more true if the revenue gained is used to fund public transit which creates a little extra draw for those on-the-fence drivers.
Those changes will also enable new possibilities in city design, which you may not consider possible today. The attractiveness of that system might surpass both your present day reality, and the less traffic clogged future, and you'd change your mind about personal vehicles.. but if not, you still have less traffic.