I suspect when driving demands attention, it's easier to buffer incoming words to handle a second later, or to simply miss portions and recover.
In contrast, a buffer of outgoing words between mind and mouth is harder to manage, and people prefer not to fall silent in awkward socially-unacceptable ways, especially if the other person might not see what diverted you.
It can, but I've heard quite plausible claims in the past [1] that you shouldn't let it - because that's one of the things that kills motorcyclists. Your autopilot brain is looking out for other cars quite effectively - but a motorcycle isn't a car, and can slip through un-noticed if you're mind is engaged elsewhere.
[1] Citation needed, but lacking I'm afraid!