No one seems to have picked up the one thing that city hall seems to be worried about, which is real, which is congestion.
In a free market the streets would be extremely full of taxis (broadly defined) hanging around hoping to be closest at hand when someone needs transport, to the point where this is a significant nuisance for everyone else. It is for this reason (at least originally) that most cities limit the number of vehicles allowed to pick up
"street hail" custom. In London (and probably elsewhere in the UK) there is a separate category of vehicles which are allowed to do pre-booked runs (originally you'd have been booking by phone from a landline, so it was really quite different). There is, apparently a phenomenon of professional Uber drivers hanging around near likely pickup points (stations, airports) to the point where it does or might cause congestion, so the same rationale as applied to taxis would argue for somehow restricting this (eg the 5 minute rule would make it more sensible to be parked rather than cruising).
An alternative solution though would be to increase the congestion charge (a generic daily charge for using central London roads) and extend its reach until congestion dropped to acceptable levels. The money could be used to reduce other taxes. Even more extremely, they could just decide the number of vehicles they like in central London and auction off that many daily (or hourly) tickets instead of having a flat charge.