Good observation. That is a TX31. The 50-series have a pulley on the rear cable adjuster barrel. This version is withOUT a claw mount. It's designed for any bike that comes WITH a derailleur hanger. Only it looks like it needs a claw bracket because it uses a relatively heavy. stamped, black, thin steel tab for the mounting plate and I don't think there is a B-screw for this one. But I've verified it works fine with 8 speeds on indexing using a 6.9mm wide chain. I didn't splurge on a 9 speed chain but the pulleys seem relatively narrow.

I'm not sure if the RD needs replacing. Some lube, and a ft of new compression-less and lined der. cable housing with a fresh cable is likely to get the rear to shift fine. However, remember to check the der. hanger alignment. If the dropout/frame is steel, then it's possible to bend it back slightly. If it's aluminum, it may be hard to find a replacement hanger, or if that tab is integrated into the dropout, like it appears, it may not be easily repairable if bent.