This is an enormous assumption (frequently wrong), and precisely why the costs of a government mandate should not be implicitly laid on a select population.
It actually results in attempts to cheat and incentivize corruption. For example, NYC had or has a problem with cabs not taking people to poorer neighborhoods. The government can mandate it all day long, but they did not stop cab drivers from discriminating. The correct solution in this case, would be to pay the cab drivers the market price for going to the poorer neighborhoods.
The payment obviously would have to come from the government, either given directly to the poorer person or can be given directly to the cab driver. But either way the incentivizes would be properly aligned, increasing supply of cab drivers willing to drive to the poorer neighborhoods.
Where this falls apart is that it requires increase in government spending, meaning increase in taxes for rich people. And obviously, they are going to oppose this wealth transfer. It is much easier and cheaper to simply require cab drivers to go to poorer neighborhoods under threat of fines or whatnot, and sit back and let the status quo continue.
It depends on the context of mediation. In one sense, government-as-mediator is completely compatible with a libertarian nightwatchman state so long as such mediation is impartial and does not infringe on public and private rights of either party.
But I don't think such a distinction helps here, as the functional differences between taxis and Uber amount to little more than legal fiction.
Instead of the government acting as either entirely public or entirely private, why shouldn’t it act instead as a mediator between the public and the private? This seems to be the most logical decision for me compared to either full socialism or full libertarianism.