In my experience the 029 struggles in oak above 15" even in perfect operating condition... I have had several, and use them as the main saw I lend out to new users & new guys on my firewood service I run in the fall. They are really robust, but not very powerful, and need to be very sharp to not easily slip the clutch with a 20'' bar in oak.
I would also recommend sharpening the chain & going easy on the rakers when you do, as I find .65 gauge to be too much off the rakers. Needs to be more like .58-.63mm for the 029 IMO. A Sharp chain and nice even rakers, and I am willing to bet this issue will not trouble you anymore!
I would go ahead with the sprocket if you wish to, but the clutch is likely in good shape judging from lack of heat discoloration in the clean rim photo.
If that's the first sprocket you've gone through, then that saw is practically brand new
Luck