Thanks for the update, sorry for the frustration.

having a big big cog shouldn't affect the ability of the derailleur to drop to the smallest cog. The only issue with too big a big cog is that the derailleur might not be able to stretch the chain far enough--it has to move in and down to get the chain around the big cog, and if the lower arm is too short it simply won't fit. Usually a little space can be gained by adjusting the B-screw. Still, the derailleur is returning to a "rest" position---untensioned---when dropping to the smallest gear. The size of the big cog shouldn't matter.

if it was me I'd look really closely at anything which might be impeding the derailleur's swing. I would probably blast WD-40 into all the derailleur joints and work them a lot in case some old grease gummed up. I would probably use some language best not used around children.

If the derailleur is thousands of miles old, its main spring might simply be worn out. i'd try the new one, and also consider the washer ... squeeze another few miles out of it maybe. I have never looked into rebuilding a derailleur, but maybe that would be a fun winter project ... for someone else, not me, but who knows?

As I type this, i have an unlocatable ticking noise in one of my bikes ... it is easier to commiserate with you than to spend another hour (probably the 20th or 30th) doing everything i can think of to find and fix that problem