The Jaguar version of Wolfenstein 3D did not use the shoulder buttons on the Pro Controller to any meaningful degree. They certainly did not strafe. I have a Jag, the game, and the Pro Controller. It's still a great version of the game, though. I tried playing the browser version, and I can't stomach it.
He might have gotten the Jaguar version confused with the SNES version. I don't know if that version has shoulder strafing, but I know SNES Doom does. (Yuck.)
In fact, the Jaguar version of Doom didn't even have shoulder strafing. I don't think any Jaguar games had it. Alien vs. Predator had it on X and Z, of all places.