A few thoughts I have:
3.5 turns seems a lot to me too but it's not critical , and if that's what the manuals say... It affects the tickover/low revs and your bad running implies deeper problems.
If it's running rich no surprise there's improvement with a mixture screw fully in.
Have you checked the diaphragms? Any hole/crack will affect pick-up. And the seal on the diaphragm cover.
Have you checked the inlet manifold(s) for cracks?
If the bike is running, spray some brake/carb cleaner in the area and see if the revs drop. If so there's an air-leak. Not necessarily the manifold of course. Maybe the choke bore?
Is there a little filter on the fuel input pipe? (like a DR)
Has a prevous owner been tinkering? Maybe an o-ring is missing, jets are wrong, needle is too low.
Worth checking a schematic to verify.
I expect a refurb kit will be a great improvement and a reassuring baseline. Apply lots of carb cleaner and let it soak. Then compressed air if you have it available.
Hopefully you'll be able to rule out the plug connectors soon.
[Edit] Maybe some heat from a soldering iron to help move the broken jet.
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