While I would have trouble measuring the time it takes for a typical incandescent automotive bulb to turn on to nearly full intensity from off, or to turn off, I can certainly see the difference looking at vehicles in traffic. In fact, with some incandescent assemblies now being designed with faceted reflectors so they look like they are composed of a number of small point light
sources (to look like the more expensive LEDs) the best way to distinguish the design is by switching time. I find it more apparent on the going-off side than the coming-on side of the cycle: the LEDs just "snap" off while the bulbs take a perceptible time to dim.
The ramp time is really quite short, even in relation to response time. I would expect any improvement to come not from a reduction of the tiny fraction of a second taken to turn on, but from the greater chance that the very sudden transition will be noticed by the other driver.
Benny has a good point, that some bulbs are better than others at handling vibration. I have used both standard and "rough service" bulbs in my ordinary "trouble light" style of work light in the garage, which gets banged around and dropped. The rough service bulbs really do last much better, but they are costly and less efficient. Still, they eventually die, and an LED would last much better. Tail lights
are just a permanently installed version of the same situation.
I think a significant problem with dirt, aside from obscuring the lens, is interfering with contacts. The advantage in dirt and water of the LED design is simply that the LEDs last so long that they are not made removable or replaceable; when they die the whole lamp is replaced. Non-removable means no contacts, which means no loss of contact, and thus reliability.