Why don’t recommendation systems create, or allow for, a way to post recommendations that respects poster privacy? For example, I may want to post something about a doctor that I use but not want to reveal my true name or information that reveals too much about me — say, an indication that I have some particular health problem.

Yes, there is a tension between anonymity and usefulness/veracity of recommendations, but that can be dealt with separately; it explains why this could be an option for some posters.