The comparison doesn't fall apart. In both cases, government regulation, in the guise of consumer protection, limits competition and increases costs. In the case of Taxis, the rates themselves are typically fixed by the government. That's why a medallion can be worth $1/4 million. Ridiculous. We can agree that there shouldn't be two sets of rules. My solution is get government out of it as much as possible for the benefit of the consumer. Uber is an absolutely great thing, especially for safety -- drunk driving deaths and serious injury are demonstrably down.
Perhaps you are unaware that their refusal is necessitated by government regulation that could give them liability. Once again, the best solution, in my opinion, is to have less governmental interference. That way, if the market wants (and is willing to pay for) background checks (and, like you, I would prefer knowing that the drivers are not criminals, especially when they are driving my daughters), Lyft will provide that if Uber doesn't. On the other hand, if market doesn't want (or won't pay extra for) background checks, who are we to insist?
Given a chance, the free market works REALLY well. Ironically, it is toughest on business owners, hugely benefits consumers, and is good for "labor" that is willing to work hard and produce. At least that is my strong belief.