In the circumstances you described, there is absolutely no way I would ride a flat tire on my nice road bike. I am almost certain there would be some damage to my rim and I’m too anal to allow for any damage. I carry a spare tube and a patch kit. If that doesn’t do the trick then the cell phone, if that doesn’t do the trick then I stop someone to make a call for me or ask that they make a call from somewhere there is a cell signal. I would walk 5 or 6 miles if necessary. If none of that worked and I couldn’t come up with some sort of temporary fix, then I would sacrifice the rim. I cannot even fathom why they did not accept any help and were willing to risk damaging their rims unless they recently won the mega millions in which case I would just have a chopper pace me.