No 'dis' intended to this Forum whatsoever, but first thing I'd do is at least consider "restoration" and the probability of whether your rifle ever had such a factory pad, which I sincerely doubt. Yet, 'if so', with possibility of "factory original", might wish to reconsider as even "leaving it alone" as collector variation and simply address the stickiness. For that info and all nuances technical to restoration, I'd contact the Winchester Collector's Association and present the situation to them for their opinions & forensic quality advice.
Based on the condition of the metal as you describe it, I'd likely go for restoration in any case. An era original stock, typically defined by "era correct butt plate" perhaps worth seeking. But butt of any era w/plate, likely better than a 'done up' pad even if Win generic correct.
Any pad acquired and installation cost to be considered against 'net' of stock & plate replacement plus whatever effort & correcting your 'sticky wicket'!
Then eBay, whatever sale value of your existing combo. All as a 'net-net' alternatie of stock & plate screwdriver presumed installation! If you do decide to to go pad replacement, both WRA above and perhaps the Montana 'everything pre '64 Model 70' guy might be a good resource for a lead as likely repro pads as likwly generic between models. Buy or simply get advised referral:
https://pre64win.com Strictly re your 'sticky wicket', that as another matter of research where I'm no help. Would be nice to have a few pix of the "surgery candidate" if convenient!
Good luck & Best!
John