Well, look at it from their perspective. They ask if you have a spare, if you don't, and they dispatch out, and it is determined they cannot repair the tire, and they do not have a tire with them, then they wasted a trip, their time, and your time.
Just because you think it leaked out the extension, doesn't confirm that is what actually happened, and just in case it is a hole you cant see, that may or may not be repaired, they would want to have a tire with them when they come out.
Now if you purchase the tire, and it is not needed, do you get a refund? That would be my question.
One thing I learned from all my travels, carry a small compressor that can go up to at least 120PSI. This way if I do get a small leak, I can reinflate the tire and hope it last me until I can get to a tire shop. I also carry a tire repair kit that has the fiber rope, tools, and rubber cement, just in case I find a nail, then I can at least plug the hole long enough for me to get it repaired correctly. I had an inner dually get a nail, was able to repair it still on the truck, never did get that tire patched, the plug held for years until I got rid of the truck.
