There is some wear on the 2-3 shift fork, it could cause some of the problems you are experiencing. The "big gear with the thin collar" is actually the synchronizer assembly. It is also referred to as a "drum". It has brass rings on both ends which help engage 2nd & 3rd gear.
It is engaged by the 2-3 shift fork moving the collar. 3rd gear is to the front. Take a close look at the ends to see if the brass rings move when it is moved manually. If the rings are loose or worn excessively it won't engage or disengage properly.
Fred
Thanks for the lesson. One of the things I thought was strange is that the rings under the assembly moved separate from the drum. As I moved the drum back and forth the edge of the brass ring was visible and had some slop in the movement. I wasn't sure if that was normal or not. Sounds like it isn't. Do you know of an exploded diagram of the gearbox, or drawings/illustrations showing how it works? When I had it apart a mechanic friend checked it out and now it's bugging him until he figures it out. 8-)