So I Googled "Tecumseh engine makes thumping sound when running" or something like that, and I came across a reference to someone who solved a motor thumping problem by tightening the bolts between the engine and the deck. Lo and behold, when I gave my engine a vigorous shake I could feel it moving a little bit. It appears to attach with only three bolts, and one of them was a few turns loose!
I just spent a couple hours scraping the worst of the goo off of it, tightening the bolts, zip tying some cracks in the shroud that covers the clutch mechanism, greasing the bearing in the clutch pulley, sharpening the blade, and cleaning/oiling the air filter. I also drained the oil and gas, since I knew it was going to be tilted for a while and would be hard to start if I didn't. The oil looked terrible, perhaps my topping it off constantly tactic isn't really working
I refilled it with Delvac 15w-30 after I got it put back together, that seemed most appropriate from my "stash". I think I might mow a few times then drain the oil again, see if it looks any less terrible when it comes out. I would think that HD oil would help clean things out to some degree…
Anyway, it looks like several parts that probably are not replaceable for a reasonable amount or possibly available at all don't have a lot of life left. The friction material in the blade brake is cracked and the rivets are loose. The spring on the clutch cable is wired to the arm, and the cup that goes between the blade bolts and blade isn't far from wearing through from rubbing on stuff. I suppose I'll use it till something breaks that I can't or don't want to replace, not sure I want to make room for two mowers so maybe I won't do anything until then...
Thanks for the suggestions, Doug