Can political involvement hurt admissions chances?

Indeed.

Advice here to act on such fears by concealing memberships/avoid pursuing it IS self-defeating as most respectable/elite colleges DON’T want students who are afraid to stand up and make a good case for their beliefs whatever they may be.

And it wouldn’t have worked on folks I know who are passionate about activism in political/religious areas…most IME would feel such advice would be asking them to go contrary to their own personality and character…to be shrinking violets when they’re actually more like proud thoughtful roaring lions, bears, and tigers as IMO, they should be with pride.