I have loaded for a Handi-rifle that belonged to my Dad. It shot considerably better than I thought it would, for such an inexpensive rifle. He had requested that I handload some ammo for him, and with him being 400 miles from me, I took some FL sized cases and loaded up some mild to medium loads (several powders behind the 63 gr Sierra SMP) and drove over to his place to test the ammo. What I found was that the medium loads were pretty hot in his rifle, resulting in flattened primers and stuck cases. So when the OP was talking about stuck cases, I immediately thought back to my experience. In that rifle, I would only FL size cases. If the OP neck sized cases, particularly if they were from hot loads (and again, medium loads in that rifle can be hot loads), they would be hard to chamber and would likely stick in the chamber before being fired.
And what you said about running the rounds into the chamber before going to the range is most certainly what he should have done, and I'll guess that he'll do it from now on.