I picked up these chairs last week  

Wow. Unbelievable find. Congrats indeed!
They still look great and with some additional elbow grease they will hopefully be as shiny as ever.

Why every home should have a splash of blue Ballistic. Nothing could be finer.
yes,
Aunt Mark

I would like to have spoken to the people who dropped them off. The seats are absolutely perfect - as if they have been kept covered - but I'm curious about what is along the top of the chair backs. People have leant on the back of the chairs over time and it's almost like a layer of wax. You can see where I have scratched a bit off. I'm wondering what sort of business environment they have been in. I'll be getting a local upholstery cleaner to deal with the fabric, but being able to guide him on what that stuff might be would have been useful.
The arms have patches where the lacquer finish has come off, but I think I'll live with that. Maybe one day I'll strip and polish them. The frames and legs are perfect.

Beauty salon.
Aunt Mark

Ah, very possibly, Mark. I hadn't thought of that one. Upmarket hair dresser. It means the odd stain on them might be hard to budge I guess because it will be hair dye.

It's permanent. Maybe you could dye the chairs?
Best,
Aunt Mark

There are few methods that shift ink and dye. I'll do a test and see what happens. I might get lucky. There are only a few stains and they're not too obvious. The thought of dismantling and dying those chairs send shivers up my spine.

Try cheap hairspray to lift the dye stain. Must be aerosol. Soak the stain. Blot with a white cloth..and see if you get anywhere. Or Folex. Or just clean 'em up and live with the stains. They are part of the chairs history. And maybe Wayne Newton sat in one, once.
Best,
Aunt Mark
EDIT Please Post Pleasant Progress Pictures.

I'll post the unpleasant ones, too.
I'm sure you are right about the previous use of those chairs - I'm thinking the almost waxy stuff along the back might a build-up of hair product.

I'll try the hair spray method and take photos.
I'm looking forward to using those chairs at our dining table. It means the bright yellow shell chairs will move into our AirBNB pad - that place is ending up with some nice furniture in it.

The crust is built up hairspray. If those chairs could talk!
Aunt Mark

If those chairs could talk, they would never shut up.

Hairspray crud should come out with alcohol, right? (Not vodka. Rubbing alcohol will do fine. Well, maybe vodka would work too, or maybe save that to drink when the other stuff doesn't work.)
Soak a q-tip in alcohol and dab the crud and see what happens. Maybe test it on the underside to make sure it doesn't alter the color of the fabric in any way but i think it will be fine.

Alcohol seems to be the popular one on google, too. I'll get some and test it - on the underside first, and then on the gunk.