Why do they send Drill Sergeants to BCTs of different MOS than theirs?

Because it is BCT, not AIT.

BCT should be the same for all MOSs (save those that engage in OSUT rather than BCT/AIT), which means BCT is MOS immaterial.

My guess is that you're specifically talking about BASIC training, right? When I returned from SE Asia in December 1968, my orders specified that I was to report to Ft Benning to be a Drill Sergeant. Like you, I was a Combat Engineer. My thoughts are that it doesn't matter much what your MOS is, to get that MOS you first had to complete Basic Training and then proceed to Advanced Individual Training. Therefore, you know how to march, run, shoot and crawl. You also know how to bellow at anyone not following your directions. That's about the bottom line.

Because BCT is branch immaterial. Barracks parties, D&C, weapons qual, ABC, NBC, CBR, etc. are all done the same way.