The 1-1/16" turned down axle is a far better design as you are not relying on a very weak lock collar to hold your axle in place. The end of that new axle is threaded and you should have a bolt that goes in the end of it. The axle usually requires a few shims between it and the bearing. The bolt in the end of the axle will hold it from moving. Tighten the set screws to keep the bearing turning with the axle, although that is rarely needed since it is tight against the bolt. Much better than the old 62mm lock collar bearings that would often break off the inner bearing race, especially on splined axles. And yes some early splined axle machines were 1-1/4" all the way to the end and used the same lock collar bearing as the round axles.