I am sure uber/lyft will be more than happy to pay minimum wage if the wage only counts when a trip begins. But the reason the companies are against the law is because the law requires minimum wage while the driver is not driving too. Which means that a driver could actually get 2 minimum wages at the same time while driving for both uber/lyft.
But of course that won’t happen. Uber/lyft will ban driving for competitor. Set up shifts such that every driver drives 37.5 hours a week. It is simply not worth the hassle of paying all these benefits for drivers who log in for an hour a day at random times.