Now you got two threads??
"" I took off the cover and can't see anything that appears to be broken. The spool turns freely when off but binds again when I re-install it.""

Sounds like the starter pulley is not mating with the flywheel as it should.

Dude : I would agree with Glenam it would appear that the recoil itself is the problem . It probably is binding , if you intend on keeping the saw it might be worth putting a few bucks into having it serviced for a back up saw . I have a few Oldies that I still use on occasion lol .

Is the rope start cover fitting flush to the case when you install it ??
When I fit the cover to the flywheel of any saw, I always pull up on the start handle several inches, several times to settle the pulley into the starter pawls correctly, as I push the cover to be flush with the case.