Wheels had hubs. Besides, they didn't stay round.
If a customer had a damaged wheel, I think giving it a spin under the traveler would be an easy to see what size it had been.
Measuring flat iron for a tire was easy: roll the wheel along it, then cut 2 or 3" longer. Bending it into a circle would shorten it. That's where the traveler came in: measure the wheel and measure the C-shaped tire before welding.