You could get clearance to cross into an approach path of an airport as long as you are doing it quickly and without delay. It happens all the time especially in cities where there's a lot of air traffic. We do it here in DFW, but we're also at lower altitudes (GA aircraft). My guess is the pilot was at the wrong altitude.
I just saw an update with Pete Hegseth who said it was a routine training mission with an experienced crew and they were on NVGs. Still , just because you're experienced doesn't mean you don't make mistakes.
What an all around sheet show.