Fine Harness classes allow running martingales, so a rein stop is probably a good idea there. These could include various Breed showing with Driving classes.
Carriage classes do not allow running martingales under ADS that I know of, nor are they allowed in Combined Driving competitions. Not seen with antique vehicle turnouts, because it would not be traditional. Not sure about old Racing carts, might have running martingales, but those vehicles are not commonly seen hitched (at places I have been) either.
Guess it would all depend on the vehicle and setting, if a running martingale would be considered traditional/correct to be included.
Running Martingales have always been more of a “gimmick” item to me. Horse has not changed when you remove them. So you are still at square one, need to get another method to convince horse to modify his issue (often too high a head position), so he changes body position while being used. Training over forced body postion with the gimmick.