recently I have had an A65 on the bench with exactly the same over-shifting problem.
When shifting up from 1st gear you ended up in the false neutral between 2nd and 3rd. Engine and g/b were manufactured in 1971 as well.
Apparently, I have been the third guy trying to sort this issue.
Initially, I had also suspected an issue with the cam plate, In that case the culprit was something else.
In the end the issue was caused by a quadrant that was haphazardly manufactured right from the start.
There are two bosses on the inner side of the inner timing cover that limit the travel of quadrant when you shift up and down.The 'stop' on the quadrant was rather short and with some wear that took place over the years on the bosses themselves, the movement was too much.
The right one was assembled, the left one was the quadrant I took out of my spares stash.
This is the situation with the ('new') quadrant loosely assembled in the inner timing cover in the shift up-position, quadrant on the boss...
On that particular bike there was an additional issue. The layshaft was shimmed wrongly, as they did not start with the standard shim parts-# 42-3079 on the timing side. Anyone had chosen to put additional shims in there and to take a thin shim from a Triumph on the drive side end.
Surely, this did not help either as the sliding gears are too far to the left side before any shifting takes place.
After swapping the quadrant and shimming the layshaft correctly, the issue was resolved.
I hope this helps.