That TD did not know what they were saying or your trainer did not understand the TD’s explanation. Not all TDs are created equal and some know the rules better than others.
There is no rule that says you are the only allowable rider at a national (non CDI) show other than the ones outlined above - various championships or apparently the qualifying classes for various rider championships, which makes sense.
When I work in the show office and we are explaining signatures/roles to people I try to break it down like this
Rider - obvious
Owner - pretty self explanatory but it can be a representative for the owner
Parent/guardian - also should be self explanatory, but the pushback I get when I tell people that it can’t just be the trainer/coach unless they are also a legal guardian is ugly sometimes.
Now to the confusing ones for many:
Trainer - is supposed to be the one responsible for the care of the horse. If you are in full service and even at the show your trainer is the one taking full care of your horse - done. But if your trainer is just instructing/coaching you - you’re actually the trainer for purposes of USEF show rules.
Coach - that is the person on the show grounds coaching you.
Someone can be both trainer and coach or just one or the other.
Dressage people tend to use the term trainer when coach is often a more accurate term.
The only thing I also caution people about is that in theory there should be no one on your horse on the showgrounds that has not signed a waiver with the show management.