This is a symptom of the core, underlying problem. The coupling between employer, employee and health care needs to be undone. Employers aren't healthcare providers, we don't need additional middle-men and bureaucracy dictating, by proxy, how people receive care and from who. Universal healthcare would let Uber and Lyft continue to do what their primary focus is, and drivers would be able to do whatever they want work-wise and not have to worry about its affect on their ability to receive healthcare at an affordable price.
The subset of politicians who are pushing for universal healthcare don't all belong to the group supporting coupling with employment. I'm not really sure where you heard that? For instance, I don't believe Medicare for All, at least in its proposed form, encourages coupling employment with healthcare, but correct me if I'm wrong.