Does "I've got a flat tire" work too (in AmE)?You can, I suppose, use "a" or "the," but the sentence will remain odd. Instead, we would say, "My car/bicycle/motorcycle has a flat tire."
There are four tires which are equally capable of going flat. "Driving" is not enough context to tell us which tire. Rooms often have only one window. Even if there is more than one window, the people who live in a house may know which window they open to get the best breeze. "Open the window" frequently has enough context to tell us specifically which window. We do say "open a window" if we don't have such context.so there lies my question. Just weird.