I'm not sure what the situation is like in France, but most Uber drivers I've seen have the Lyft app open at the same time. It seems pretty trivial to download another app and use both at the same time (easier than almost any other freelancer finding more sources for work).
I mean, imagine if a McDonalds worker could just say "I don't feel like working for $10 right now, I'm going home!", and then just leave in the middle of a shift. Then the next day he sees that Burger king pays more, so he goes and work for them that day for a couple of hours, after which McDonalds increased their rates so he goes back there and work a few hours. Sounds like insanity to you? Well yeah, because it is insanity if employees could act like that!
Uber drivers are not employees, end of story. Uber doesn't have any power over them as is, normal workers have a contract which is worth a lot to them, you have no such contract with Uber as it is trivial to get the permit to drive for Uber, Uber drivers are not afraid of Uber suddenly firing them, they can go in and out of their role how much they want.
At least in California, driver pay, and what the customer paid are decoupled. There is no more "surge" per say, it's just a dynamic price for the user, which might be completely different than the one the driver has.
In Europe they still operate at a standard of 25% cut.
> Then the next day he sees that Burger king pays more
Why would it be in the interest of Burger King to do that?
Where I drove for Uber, there were five platforms. But Uber had 90%+ of the rides. In a year, I think I got one Lyft request, and nothing from any of the others.
If there were plenty of customers on all platforms your statement might be valid, but that's not the reality.
Generally speaking, Lyft costs more than Uber, but you get a better class of passengers and a better class of drivers.
Prices for the ride are always changing. Uber usually has lower per mile cost, while Lyft has a bit lower minute cost. This does not affect switching costs.