It's really not just a norm that office workers assume most employees will be full-time. Many aspects of labor law are set up with the assumption that full-time is the normal state of affairs and part-time work is for exceptional or temporary situations. And legally many benefits are tied to full-time status. If you are part-time and get laid off you might not be able to collect unemployment, depending how few hours you were working. Under the ACA companies only need to offer healthcare options to full-time employees.

The bigger point though is there would still be major limitations on flexibility even if Uber/Lyft was able to offer every driver the option of full or part-time. They still wouldn't be able to set their own hours, only work a few weeks here and there with no predictable schedule, sign up and start working immediately with minimal overhead, etc.