FP1 thowing out of gear- can you guys help me sort this out?

I have a mid sixties lever FP1 which has suddenly started to throw out of gear as it is starting.
I have the mill on a drive with soft start- when set in one of the higher gears, it will just throw out of gear as it starts to wind up to speed.


This is the lever position where I have the problem:



I pulled the lever cover carefully and this is the gear position found:



I see where the gear is not indexed:



Also note in this pic that the gear is not aligned well with the gear under.

Looking at this I am starting to believe that the leftmost gear cluster on the idler shaft is shifted to the left and causing the problem- I think that perhaps there is a thrust washer which is worn on this shaft.
I also can shift this cluster back and forth by a goodly amount (maybe 1/8"..).

So in looking the work over I see that the gear shift lever in question has been worked on over time:



I had seen this work as I set up the machine and wondered why the work was needed as the indexing of lever to shaft should not need changing I would think.

The shaft is at this time only indexed by a set screw drilled through the face and the taper pin is missing.
I pull the lever and look over the mess:



I am pondering this when a mechanic friend comes into the shop.
He agrees that the idler shaft will probably have a thrust washer in place & offers up a decent explanation for all that I am seeing

He thinks that the gear cluster is in fact floating too much side to side and indicates a worn out washer.
He also thinks that in service that the mill started to throw out of gear as the washer wore out.
What he thinks happened is that repairers simply rotated the lever to "chase" the shifting gear cluster as it shifted further and further.
He thinks they took this route instead of pulling the shaft and replacing the thrust washer....

So here I sit- I am in the middle of a job and the mill is down.
Is there a thrust washer in place on this gear cluster?
Will wear on this washer cause the mill to throw out of gear?

How does one pull this shaft if this is needed?
The original taper hole is still present in the shaft so I am thinking to clean up/set new pin after replacing this hypothetical thrust washer if it in fact exists..

Thanks all.
Sorry for the rambling post- long week..

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