Ideally it should be across corporations. I know a ton of people who drive for both Uber and Lyft, depending on who pays better. Thats a critical element - there needs to be a market of companies for people to work for, but they still need solid benefits and protections.
Think of this as a next generation union type structure.
I don't know why this doesn't come up more. People get hung up on whether contractor or FTE fits gig workers better, without ever suggesting there's a categorization issue here that could best be solved with a new category.
Why do you believe no taxi drivers want to be employees? Why is it different from pizza deliverers or restaurant servers as regards wanting to be an independent contractor?