I had a similar idea a while ago when I had two of my tyres slashed. I was carrying a viable spare, but with two tyres out of action, it didn't do me a lot of good.
I got a tow to the nearest tyre place probably within a kilometre, but they had to order in the tyres to match what was already on the car, so I was left without a car for the remainder of the day and some of the next one.
I thought it'd be a great business idea to be able to call up for roadside tyre replacement, similar to how you can "holler for a Marshall" when you get a flat battery.
Incidentally, I've clearly been living here too long. I had to battle with myself to not write "tire" all the time.