For the purposes of this discussion, no, the tire is not round. When considering your argument for 2*Pi*r the tire is absolutely not round. Let's do the math:

Take for example my 35" tire. Assume for discussion that I have inflated my tire to 40psi and it is a nearly perfect approximation of round. The circumference of the tire is, as we know from 2nd grade, 35 * Pi. This is approximately 110"

Now let's take my 35" tire and air it down to 10psi. Based on experience the tire is probably now around 31" tall when resting on the ground. 31 * Pi is approximately 97". Did the tire actually shrink along its circumference from 110" to 97"?

Tour assertion above is that "look, the diameter drops so of course the circumference went away." You are effectively arguing that the tire stretches along its circumference by 13" when it is aired up from 10psi.

The tire is not round, especially when aired down.

The only way to win this argument is to show that the tire does or does not stretch along its circumference when it is aired up or down. Considering that the amount of stretch we could expect here is probably on the order of 1/1000 of an inch this is probably next to impossible for one of us to prove empirically. Further, neither of us have the experience to throw out plausible theories.

However, while we're talking about thousandths of an inch consider what happens with temperature change. With a simple change in temperature from say 50 degrees to 90 degrees the tire will indeed expand some small amount, again probably on the order of thousands of an inch.

In the end its important to note that no one gives a fuck about thousands of an inch: we're not building the damn space shuttle.