Do the CRC jobs you're interested in require more than a Bachelor's? I can only speak for my own experience, but I have a B.S. in psychology (graduating with MHA next spring) and was hired as a clinical research coordinator in psychiatry with about 9 month previous experience as a research assistant. I'm still working in clinical research, but have moved up and out from that role. In reviewing and interviewing for my replacement, individuals with Ph.D.s did apply, but were often considered over-qualified. At my institution, PhDs are more often in the lab/working behind the scenes (basically everything you mentioned in your 2nd post) while the scope of the CRC role is more focused on regulatory compliance, consenting patients, and following them for study participation (level of involvement varies by study).
If you feel like you're being overlooked for CRC positions it may just be there are people applying who have more specific experiences (protocol development, consenting/doing assessments with patients, study start up, etc)? You may already do that as an RA, but I'm not sure. My first PI had me do some of those tasks as an RA because (I found out later) she was hoping to hire me as a CRC. I moved to another state post-undergrad, but it was much appreciated work experience.
TL;DR: If you'd like to be a CRC then (to my knowledge) there are many positions that only require a Bachelor's. If you already think you're not interested in conducting your own research, going through a PhD program may not be a worthwhile investment. MPH could be helpful, but probably more for epidemiology vs your previous mention of psych/neuroscience. You could also look into a masters in clinical research and see if you think that would be helpful?
Source since this is literally my first post: 9 months RA in Neurology, 2 years CRC in Psychiatry/Psychology at major medical institution, 2 years research program coordinator in Neurosurgery/Pain Med at same institution with just my B.S.