I can understand the feelins of the OP.
My DC's are now 8 and 5yrs and I have always, and still, felt slightly sick when I know they are out and about somewhere being driven by someone other than myself.
This includes my Dh, my parents and other parents.
I also sometimes stop and think about how dangerous just going in a car is and feel sick that I ever let them go in a car, even with me.
However, they do go in cars, with me and with others, otherwise thier lives and mine would be severly curtailed. There are thousands of examples of things they, or me, couln't do if I never allowed this, playdates being just one of them.
I want them, and me, to have intersting, varied, full and fulfilling lives and experiences and relationships separate from those with me.
So I swallow my fears, try to rationalise my anxiety, ignore my sick feelings to allow my sons to experience a carefree childhood.
I think you need to do some of this DD. High levels of anxiousness do not equate to good mothering as you seem to suggest, but say alot about your belief behnd this that 'I am only a good mother if I protect them at all times' you need to start to also belive that 'a good mother does not stifle her children to control her own anxiety' and realise you cannot protect them every minute of every day and sometimes you have to trust other adults to do this for you.
I understand the anxiety, I really do, but that is your emotion, your problem, control it and don't make it your childrens problem.