Non-team variants for four or more players in Mille Bornes?
This happened in my gaming group as well, we fixed it simply by having a remedy 'timer' which is once you've been stuck for 3 turns, any GO card would suffice to move again.  Likewise, if stopped for three turns, you could simply play mileage cards.

The other variant we tried, which you might like is that remedy cards (and go cards) were played into a face-up draw stack, and instead of drawing normally, you could draw from any used card in the stack, so once someone had fixed their flat tire, anyone could spend the turn to draw the remedy the second time.