Who don't think they have a problem because they can't remember? My mother doesn't seem to think she has a problem with her memory or with her driving because she can't remember that she's even been to several doctors who have told her that she has dementia and that she shouldn't be driving.
I agree that there comes a time when people need to give up the car keys, but they need to have the reassurance that life will go on. It's a really scary thing, giving up your independence. It's really hard to have to call people and ask them for rides, so maybe your friends can call her every couple of days to just ask her if she wants to get out. Just my opinion.
My suggestion is to begin putting that plan in place right now. Don't make an issue of her driving or not driving. Just tell her you'll pick her up to take her to X, or friend will be over to get a hamburger on Tuesday or whatever - you get the picture. Pretty soon, she will be used to being driven around and may not put up much of a fight. It may not happen overnight, but you and all the other designated drivers will have a chance to work out the bugs.
Even 7 years after moving in with us because of her inability to make rational and safe decisions, Mom doesn't believe she has any problems whatsoever. Even though she has arthritis and osteoporosis so bad she can't stand up straight or walk more than a few steps unaided, walkers are for "old people". She's 93. Dementia is a dual edged sword. It's like anesthesia. You have pain but you don't know it or care!