On not burning bridges with the non-paying venues:
If they pay anyone at all, ask to receive some amount of payment. You’ve proven your skill and reliability to them. If they are unable to pay and you have a regular column, inform them that you’re no longer writing for free but will do another couple of columns – along the lines of giving notice at a job. In that way, you leave on good terms. If they ever make it to being able to pay, you won’t have burned any bridges.
To the doctor writing the column for a university publication, I always thought doctors (lawyers, mechanics, whatever) did that kind of thing to bring publicity to their practice/business. No harm in looking farther afield for paying assignments, though.