Theres nothing in the primary that has to do with the shift assembly.
The problem is either the shift arm bushings in the front of the case or the shift pawl/spring in tranny.
If the shifter is flopping around, check the shift linkage and the shift arm splines/bolts but could still be an internal problem in the tranny.
Another check would be to remove the shift linkage at the verticle shift arm that comes out of the tranny.If you move the lever back and forth,the arm should have a firm pressure to return it to center.

 

I would still bet it is bushing. Mine did the exact same thing. The problem was the bushing.

 

i fixed this a while ago but figured i would put the solution for anyone searching for it. the foot control assembly i tapped out lightly with a hammer and cleaned out the opening. the control piece that the pegs attach to had a bit of rust on it, so i sanded it down with some 800 grit paper and put some tranny oil on it. i also took the linkage off and let it sit in some tranny oil too to penetrate the bushings.

so far it looks like it's fixed, haven't had the issue come back.

 

The shift shaft is most likely rusted and corroded.

Remove the shifter rod on the front off the front of the case (you will need a 3/16 allen socket or allen socket), clean off the rust from the shaft with a scotchbrite pad, lightly lube with a waterproof marine "white lube" . This will cure 95% of these issues. I have done mine several times over the years, shifts smooth as silk.

 

I've had a broken spring on my bike and I could carry on riding the bike, but had to move the lever so the detent engaged ready for the next change. So I don't think that is your problem.

Several have pointed out which part you need to lubricate! It is the pivot pin in the front of the primary casing above your left footboard, no need to take the cover off.