In a flat tire situation, if calling doesn't work, try Twitter as well, and if all else fails still get to the airport as soon as possible. The airport agents usually have a lot more leeway (and awareness) of local issues than someone on a phone sitting on the other side of the country. I've had times where the phone agents couldn't do anything so I went to the airport and the agents there went out of their way to find me a solution - even once getting together with the airline next door to help me get where I needed as all the later flights were oversold.
As said above, the flat tire rule is unofficial. You have to approach it as if you are asking for help, not demanding they do something they aren't required to do.