if you do need them you're not going to have time to go get them as an Emergency Responder. I think the pragmatic course then is to disregard etiquette and tradition for safety and practicality in a general sense.
Everyone except the driver has ample time to swap boots during the drive to the location. (In cases where not, I'd also default to robust safety boots.)
Depending on the boots, you might not want to wear them while "idle"¹ in any case, though that's mostly a non-concern with modern safety shoes.
¹ mandatory note: most emergency responders have tons of work while "idle", I just mean "not actively responding"
Everyone except the driver has ample time to swap boots during the drive to the location. (In cases where not, I'd also default to robust safety boots.)
Depending on the boots, you might not want to wear them while "idle"¹ in any case, though that's mostly a non-concern with modern safety shoes.
¹ mandatory note: most emergency responders have tons of work while "idle", I just mean "not actively responding"