Hi,
It's been a long while since I have done on of those decks, but I have worked on many of that mechanism. I'm not too sure on the spring you mention, a picture will help here.
What I am thinking though is that it is not actually this spring but indeed a clutch in the auto-stop mechanism. This (from memory) is driven off the rotating take-up spindle. The clutch makes a rocking arm push against a cam that moves back and forth. When this rocking arm does not have drive, it sits in the middle and hits an auto-stop trip pin, making the deck stop. To find this you will have to look and study how the deck works, but it is fairly straight-forward.
When this clutch pad wears, it no longer provides enough torque to push this cam onto one side, causing the autostop to trip.
Back in the day you could easily buy kits of clutch pad material, pre-cut in round shapes with holes. These days you will probably have to make your own. Don't be tempted to make it much thicker than what is already there, it just has to be less shiny. You may have some success with roughing up the felt pad.
Not sure what material I would recommend these days for replacement, the self-adhesive stuff for the bottom of furniture is way too thick, but something along that track will be what you need.
I can access my decks, but photographing them to show you what I mean could be quite difficult.
Cheers
Tony