I do carry a folding tire, I forget who makes it, I haven't had to use i since getting it.
One can easily fold a metal bead tire to 1/3rd it's normal diameter, though it still takes up quite a bit of room. I actually found it good as a tie down system. Being rubber it grabs stuff that otherwise might shift off the top of my rear rack. If you don't know how to fold a tire, it is the same as a bandsaw blade 2 hand system for which there are many videos. Or the same as folding all those metal band products that self deploy, then refold, from tents, to laundry hampers, to sunshades