youtube.com/shorts/BFqbL8e4tkA?feature=share
What should I do about this cush drive issue? Does it just need new rubber or something else? How bad is this to ride around?

1971 Yamaha AT1 fully restored. 

You didn't show the outside...  is the snap ring in place?

Everything is tight early in its life, but with use things seem to loosen up.

I bought these ring shims from McMaster-Carr: www.mcmaster.com/catalog/97022a409  and installed one behind the circlip to reduce the play.  I only needed one, but they came in a pack of 5.

The inside diameter (ID) of the ring shim is only 2 inches, but the OD of the hub is 52 MM, a little larger.  I ground or filed the inside of the ring shim until it fit on the hub. Then I installed the snap ring.

 

 

Gotta be careful shimming them up too tight but won't be a problem if its straight side movement. They mostly wear on the cast hub extension so shimming puts load on the circlip as the sprocket tries to twist--sort of--rather than straight forward & back inline movement. Check the circlip too as they wear above the groove with constant movement for 50 years--wear where the sprocket carrier rubs on it. Grab the sprocket like you grab a steering wheel & see how much rocking it has. My AT1 looked terrible with the big farm sprocket but never lost the chain with the guide keeping it on. Nowdays as a 175 & a very small sprocket it doesn't look so bad haha.

 

Looks like the circlip is there. 

What is everyone doing to fix this type of issue?

I guess no easy cure. New hub & sprocket carrier!!. The idea was doomed from the start. First DT1 had it right with the sprocket carrier running on its own bearing & was all downhill from there. Could measure the wear on both hub & carrier & calculate the difference then go get something to fill the gap. Whatever you roll up out of shim sheet its gonna be fiddly to hold it around the hub & feed the sprocket carrier over it. First bit of the hub out past the circlip is unworn & bigger of course. Maybe a "Speedy Sleeve" come to think of it, split it & fit over. I've never used one but would be great if they come in dimensions you need although i gather they're very thin--or do they come in various thicknesses? anyone?--i don't know.

Had this problem back in the 80's, but parts were cheaper then.
Bite the bullet and get this..

NOS YAMAHA DAMPER AT1M DT125C AT3 DT175B 248-25364-00 OEM Buy It Now

Be interesting to know the inside diameter of that new one compared to yours plus original outside diameter of a new hub. Wonder if that info is out there somewhere.

William , the part number248-25364-00-00 is the same as stock.
Shims will work for a little while. I made one outta a soda can I cut open and flattened. ( mine was a 69 AT1M )
But after a bit, the wear on riding it that way...the rubber just gives out .
I.E - the sprocket will spin through the gears.. and the rear wheel will not.
And we all know , pushing a bike is no FUN !
Got one of those parts and never had that problem again.

"Got one of those parts and never had that problem again."

I agree. I put one on a 69 AT1-E to replace the one the PO had siliconed in place to make sure it never moved, By Jove.
No problems yet. Well, cleaning out all the silicone from the hub was a bitch, but since then, I mean...