If this was a T5B departure, then I don't think the car would make much difference. Domestic to T5B is what, 10 minutes? By the time you go around the one-way road system for T5 and navigate the relevant staircases I'm not sure a car would save much time, and in any event, with final boarding at 20 minutes, you weren't in much of a risk there. They are also reluctant to offload passengers until 10 minutes to departure, with the notable exception of bus gates, where they really do hold to the 20 minutes deadline.

Incidentally, the Jag service doesn't save much time for T5 to T3 either. I can normally get to the lounges in T3 within 25 minutes of domestic arrival, and the car service might, might, save about 5 minutes on that. The road section takes more or less the same, it's just the security scan which is saved, plus waiting for the next bus departure from T5, but that's usually just a few minutes. With the new Flight Connection centre at T3 you have a slightly longer walk from the car drop off location close to the previous Flight Connection location. There's not a lot in it really.