I'm not sure what "TM" or "FM" you read but I was on active duty in the US Army(MPs). In training & during my 4 years in service, I
never heard any cadre or instructors say to keep the trigger(index) finger on the trigger.
We were taught to keep sidearms condition three(03) which means; empty chamber, loaded magazine, safety on.
You put your finger on the trigger guard or frame until you were ready to shoot.
That's not always practical but it is very safe.
I for one, wouldn't go by some cadre or ROTC trainer who wants to go "rogue" or thinks he/she is some DevGru or ACE operator
.
Learn the
basics first. Then advance to the other courses/tactical training.