These are per-city rules in much of the US, but the limo companies that Uber was originally subcontracting to had to meet similar requirements in San Francisco as what tou describe. Booking is just the test that was used to determine which set of regulations apply to any given compan. For example, taxi supply was artifically constrained by a medallion system, but there was no such limit on the number of limos allowed to operate.
(Past tense b/c I don’t live there anymore and am not up-to-date with current developments)
They certainly have used the loophole you describe extensively, but that started later, once they were sure there were no repurcussions for the limo dodge.
Uber didn't only become big because of the success in the US. When they scaled their operations, they had enough capital to disregard local laws (Berlin comes to mind), where they simply paid the fines with the capital it had raised, allowing to create a brand and household name. Thats what allowed them to become common everywhere; now that they are regulated, they have quasi no advantage over any ride sharing (even taxi) companies.
This is why fines should increase for repeat offenders to the point where they quickly become unmaintainable as "cost of doing business". They should also always be higher than the estimated profit generated by the offending behavior.
(Past tense b/c I don’t live there anymore and am not up-to-date with current developments)
They certainly have used the loophole you describe extensively, but that started later, once they were sure there were no repurcussions for the limo dodge.