Uber and Lyft offers your ride to drivers and if they don't accept, then you're SOL. The time estimate is based upon the closest driver accepting your ride request, which is not guaranteed. Lyft will auto assign rides to drivers by default, but drivers who have driven for Lyft more than a day will find the toggle to switch it to manual accept ot deny. Uber's higher ranked drivers will get your origin, rough neighborhood of the destination, distance for the ride, and how much they'll make. Most drivers won't take low fare rides unless it's a good fare to distance ratio.

As for the gyrations, that is on purpose. It purposefully cycles through your origin and destinaton so that if you're screen recording for the purpose of showing others how bad their system is, you're exposing your origin and destination (because even with video editing, it's difficult to block that all out). Uber and Lyft do not operate honest businesses.