Uber is a non-profit version of Uber. They made a 10% loss last year. So a non-profit (assuming they intend to at least break even) would actually need to pay the drivers less.
I think the intent of the parent's question was to say why can't a group of drivers get together and create a ride sharing app that is jointly shared by all the drivers and just gives all the money to the drivers, hence taking no profit (rather than the more formal term 'non-profit' which has the tax implications like you say.)
Besides, companies only pay taxes on profits, so loss making companies don't pay any tax either. (And usually have some kind of ability to carry forward losses to reduce future tax bills)
Why would anyone work voluntarily for what would essentially just be a company that gave all it's profits to the drivers but expected them to work for free. Volunteers tend to work for causes they believe in for some ethical or moral reason, and while I sympathize with the drivers like I'm sure many do I doubt many would work for free so someone else can earn more.
Why would some people work for free so that drivers can earn more or riders could pay less?
Uber racked up a $20 billion cumulative deficit by subsidizing both drivers and riders.
Even if you cloned the Uber app and had people volunteer to do customer support, you'd still wind up with a service where riders pay more, or drivers earn less, or both.
That works for BlaBlaCar [0] that's true carpooling, without drivers trying to make a living on the platform. Riders pay less and drivers get paid less.
But if Uber is losing $4 billion a year at the current prices, it's hard to see how there would be a sustainable market where drivers make more and riders pay less.