ichiwowa, there has to be a way to locate the axle in the fore and aft direction, especially to cope with the backward force on a wheel when the tractor hits a bump. That could be done in various ways, but using the axle-tilt-limiters is a possible way as long as the slotted plate is rigid enough for the purpose. It would be best if the (slotted) support plate is behind the axle, since the wheel-bump load would nearly always be backwards. If the steering has a fore-and-aft drag link from the steering box on the bottom of the steering column, that also produces some fore and aft force on the end of the beam axle. I think as far as tie rod (i.e. lateral) steering forces are concerned, I'd rather make the central pivot of the beam axle big enough and strong enough to resist them unassisted. Multiple redundant restraints are seldom reliable because they each have different rigidities, and you never know quite how much load each of them is taking.
So, at this point I don't know if there is something else resting against the back of the axle to locate it along the tractor axis. If there is nothing else locating the axle, the slotted support plate has to do the job. However if that plate is behind the axle, it may not be necessary for the pins to have nuts on them: the axle is fairly strong in bending, so supporting both sides from behind should be sufficient.
Back in the days when cars had beam axles on the front, they used to have rather strong axial location measures, but the biggest forces they had to cope with were caused by the front brakes ("spiking" the brakes causes very large forces on the axle due to suddenly absorbing the rotational momentum of the wheels and tyres). However ride-on mowers seldom have front brakes, and besides that they have tiny wheels and tyres, and travel at very low speeds, so wheel and tyre momentum is not a factor.
