Well it seems that the drawer mechanism is very sensitive to placement... On an irregular surface, or a slightly bent shelf the base plate will be askew and this partially blocks the mechanism on my example.
I guess something is wrong.
The tape transport was "serviced" with new belts at some point.
It was also literally spread with some sticky silicone grease, which seems like a bad idea.
The rubber on the two spools was dry and hard, and the pinch roller glazed. I will need a new pinch roller, but I had proper tyres in my box of tricks.
All functions work after a few new caps and solders—there were a few dry joints. Recording works fine, but everything will be properly aligned when I have time (especially the bias and level settings as always).
Any more bits of advice from fellow Beocord 5000 owners?
Thanks!