Karl, this reminds me of an issue I had with my new Trunnion/spindle assemblies from Moss. As you pointed out, the new ones have no cotter key hole in the spindles. I decided to use the original spindles, with the holes, in the new trunnions. The old spindles pressed right out of the old trunnions, but I could not get the new spindles to budge in the new trunnions.
I mention this because part of the propper assembly is to ensure the spindles are centered in the trunnions. If they are substantially off center, you can run into the issue you describe, where the nut hits bottom on one side of the spindle, but the bearing assembly is still loose. the corollary to this is the short side may not have enough thread to capture the nyloc.
This is easy to check by measuring the spindle length on each side of the trunnion. As Randall pointed out, the usuall solution is to just use the bearing nut to pull the spindle into center, and then continue on with setting the correct end play. But! I know from experience that the he trunnion/spindle assemblies may be to tight to be able to center the spindle if it's off. You may have to exchange it with your supplier.
I had to exchange my trunnions with Moss 4 times before I got 2 that were good. One had an off center spindle that would not move. The other 3 had loose threading for the vertical link.