Good news! I was able to fix the problem for now, thanks to the tip from Rhino. It's not perfect, but it works.
The only metal part small enough to fit that I could find was too obvious: those damn things that come with every cables...
The metal inside the plastic is very thin and easy to put in shape, but solid enough for the purpose intended. I could not put it inside the whole spring, because the two main parts where too much separated, but it help a lot to straighten one bit while I used a pair of pliers to give the whole thing a better shape. The area is really too narrow to work easily, but it finally went the good way!
This really helped me to understand how the synth worked. And since it needed a good clean-up, I didn't mind taking extra time playing in its guts (I'll post pictures in the appropriate section). Here's a look at the dust & stuff I had to remove (you can see the before & after on each):
There's only some broken sliders to repair now, but it's really out of my league so I'll leave them that way. They're working, so it's good enough!