Originally Posted by FBinNY

Sorry, but you have this backward for the normal tire and rim configuration.

Tires are pinched in forming sort of an Omega shaped profile. Air pressure tends to make the internal chamber round, wxcept where constrained by the rim. Allowing the beads to move apart increases the working circumference by the change in rim width (circumference = flat width of the unrestricted wall of the tire + rim width). A bigger circumference means a bigger diameter, both vertical and horizontal.

This would hold true until the rim was so wide that the tire ws stretched into a low arch, in which your logic would hold. But tires aren't on rims that wide, so it doesn't.