I think the clause "for military candidates" is thrown in there because it might be hard for military candidates to find a PE teacher or liaison officer (since military candidates aren't directly assigned one). So the clause is thrown in there to make it easier for military candidates to complete the CFA, not necessarily to restrict a commissioned officer to conduct a CFA for a civilian candidate. I am sure if you ask USNA, they would say it is okay.
I think the problem with the coaches (and please don't quote me) is that they ran into some "fairness" issues -- i.e. the candidate's own coach administered it and aided them. Yes, there is a loophole in that if your own coach is a P.E. teacher, they are allowed to administer it. I don't think it has anything to do with "PE teacher" qualifications...otherwise BGOs, officers, etc. wouldn't be able to conduct it. I am not so against having BGOs and/or commissioned officer or CPOs administer the exams...it could take a lot of bias out, but at the expense of placing a heavier burden on BGOs and further restricting who could administer the exam.