The tension is real and you should not pretend otherwise
Every piece of job-hunting advice tells you to have a clear, professional, recent photograph on your profile, and that advice is correct. Profiles with a good headshot get looked at. Removing your photo to reduce biometric exposure costs you something measurable in exchange for something diffuse.
So the useful question is not whether to have a public photograph. It is what that specific file should be, given that it will be crawled, indexed and matchable for as long as it is up, and probably for a while after you take it down.