I think one of the most important parts of the page you linked should be highlighted:

WARNING. This shoulder bolt has a nut on the back side. If this nut is not removed, and a big screwdriver is put on the shoulder bolt to turn it, the shoulder bolt will shear. This renders the shoulder bolt useless. If this happens nothing short of ordering a new shoulder bolt from Pinball Resource (part# a1059) will fix the problem. So the moral of this story is to first remove the nut from the shoulder bolt before removing/adjusting this bolt. (And even then during nut removal sometimes the shoulder bolt still shears.)

Before disassembling the whole unit, I would loosen that shoulder bolt, clean the pivot point (http://www.pinrepair.com/em/gtbcred1.jpg), etc and make sure the lever moves freely. I've found a lot of steppers that were completely non-functional only needed a bit of care at that one point.

If you are stuck for a replacement, in the past I have machined down a case screw from a PC using a drill and a file. Not pretty, but did the job.

20pcs-Desktop-Computer-PC-Case-Thumbscrews-Thumb-Screw-6-32-Black-Metal (resized).jpg

You're not the first, and won't be the last. You may also have found it sheered off anyway. I'm sitting at 50/50 over 2 machines on that nut sheering the bolt.

Here's a pic of the quick replacement I made from an old PC case thumb screw:
16123238_268544953558420_436900583203405824_n (resized).jpg
The shoulder was marginally too big so I put it in the drill chuck and filed it down a bit.

Whilst slightly overkill, this thing is going to end up better than new. No grease to go hard and gum things up for a start.

It's just cotton isn't it? Could you use a round shoe lace? Otherwise I'd be looking for spaghetti insulation like this maybe? https://www.tubesandmore.com/products/spaghetti-old-style-white-20-awg