I think it's that shaft shown in the drawing above with the nut on the end that tends to seize up with corrosion and cause stiffness in neutral
It is indeed ... inside the shaft there is a pair of very light plastic bushes which only go in about 25% from each end, and the shaft seems to be factory dry-assembled (bad), so the gap between the bushes fills with electrolytic crud.
Spraying the area or bushes with WD40, etc tends to make the problem worse as the the crud swells up and grips the bushes even tighter.
However, the 2 bushes can be bought separately from Fiat @ about £1 each (you MAY have to buy a pack of 5).
The fix is ...
loosen the nut (leaving in place for tapping),
tap on the nut end WHILST "exercising" the balljoint-arm back & forth with an adjustable spanner set to fit the arm (it's usually WELL seized), & at some point you'll need to remove the nut and swap over to using a drift on the spindle (= LIGHT taps + MUCH exercising).
When you've worked the spindle free ... pull out the old bushes with long nosed pliers (easy), de-crud & clean up the spindle & shaft, and reassemble with generous copper ease + new bushes.
Finally, (after also having done the "cable clean up" you mentioned) try out & play with your sweetly responsive gearchange .
.... PS .... make sure you have Madonna's "Like a Virgin" playing in the background
..... PPS ..... When you've finished, you MAY get an urge for a cigarette and/or drink
(this is slightly is slightly perverted, but probably normal )