Unless Lyft didn't charge for rides then regardless of how they describe it, fundamentally it is still a pay-per-ride service (i.e. taxi) and still subject to the taxi industry regulations (which I don't necessarily agree with but exist nonetheless).

I 'm curious if companies like these could legally operate by not charging by time or distance but instead operating as a private club wherein someone purchases a membership that grants them free time or mileage (up to a certain limit) or some system based on pricing sort of like how NYC MetroCards are set up).

They didn't "charge" for rides for a long time. They called them "donations" and you were perfectly entitled not to "donate" anything. Of course the driver would give you a bad rating if you "donated" anything less than the suggested "donation."

Lyft even called the fares "donations" until roughly 6 months ago.

Source? Or is this conjecture on your part?