I assume you’re talking about paid depts & not volunteers but that answer will vary by department but probably not. How far from the firehouse do they live? (Latest best practice is to not keep your gear with you in your car as the gear can absorb smoke & then off-gas dangerous chemicals. Yes, gear can be cleaned but typically not after every job). How far from home are they when the call comes in? What do they do with their kid, especially if they’re the only caregiver at the moment?
Remember, they’re in their own cars, travel time is that of a regular citizen, stopping at all red lights, etc. When do they go on the clock, when they’re sitting in traffic or only when they get to the station? (Most people aren’t paid for their commute). Every FD uses some accountability system to ensure you know who’s there & who’s in the red/danger zone; The manpower officer probably doesn’t want to deal with singles checking in with him, who then need to be paired up with someone to make a team.