Most karts have a plastic piece that holds the top of the steering column at the steering uprights. I feel that it is not ideal, but I have seen a couple of people take a large vice grip and squeeze the plastic to hold the steering shaft in place during alignment.
There are devices sold which lock the steering via insertion of a pin. Most of the ones for sale are for “American” karts, but some advertise as being for the 20mm steering shaft common to Euro karts. There is one listed on eBay that attaches to the pan, and locks the shaft by holding the tie-rod flange.
Note that the position of the steering wheel itself is not a good indicator of Centered Steering, which is essential to a proper alignment. The inner tie-rod attachment heim joint locating bolts should be equally spaced from the center line of the steering shaft when viewed from a front/overhead position. Many (most) karts have a steering shaft position which has an axis that is not parallel to the center line of the chassis, going from center at the bottom to left at the top for obvious reasons of driver placement. Centering the inner rod ends to the shaft instead of to the chassis is important, IMO.
Critical from a functional standpoint is understanding that the inner rod ends are moved in an arc, and one with a much shorter radius than the spindle arms are similarly moved. If those short arcs are not equal side-to-side, then the kart will behave differently in right and left steering events.
A note about your tire wear: Depending on the number and severity of turns in each direction per lap, one front tire can see more friction load than the other. The single most damaging (tire wear inducing) adjustment is toe. Combine that with a slight mis-adjustment of camber, and a front tire can wear really unevenly.
One thing to check: Pull the kingpin bolt on the “bad” side to see whether it is still straight. Even a slight bend or wave will cause the problem you describe. Of course, observe the kingpin bearings too.