As far as embarrassment goes, my kids beg me to come eat lunch with them at school or to volunteer for field trips or whatever when they ask for parent volunteers.
I don't get the whole parental embarrassment thing. My mother always dropped me off right in front of the school every day until my senior year in high school and I always kissed her before getting out of the car. Kids that are embarrassed don't know what it is like to lose a parent.
They never said they were embarrassed but i felt they were too old for me to stand at the bus stop with them, especially with my son who was 6' 3 by then.