No, not everyone does. We were warned when it was hot not to leave them in the car, so we never did put them back. I usually carry in my purse, but often I go bagless and sometimes forget to put it in a pocket. It's also stupid cold here and I am not taking my gloves off to get my hands wet and have them freeze more. The mask can really wait!
Plus I really prefer to wash my darn hands properly to remove the mask as that is best.
I personally cannot believe people actually care and judge people for doing it. Like I said here, my mask keeps me warm and I keep it on for that reason too while driving. Edit: I'm half tempted to wear masks more in the winter next year because they are better than any scarf I've had