I don't think I'd assume the tire "went bad" due to age or anything else really other than something it hit. There's no reason I wouldn't trust the other 3 for at least the rest of the trip, other than the fact that I would not have a spare anymore so another flat would be bad news.
As for getting people to move, maybe turning the hazard lights back off and using just a turn signal would better communicate your needs. Then, if that doesn't work do as the truckers do and just start moving over. You're bigger than them. ;)
Bummer about all the damage but as you said at least you're all OK.
