I have a few of those Tecumesh motors here but I haven't bothered spending the $20 odd on a carby just to see if they will run. Not much point in spending $20 on a motor that might be worth zip
You do know a good ultrasonic cleaning, a float valve seat, bowl gasket, and a couple o-rings is about all you ever need to do these; unless, the primer bulb is bad. And for test purposes only the float valve seat and the two o-rings on the nozzle after cleaning is all you need to test.
As for compression these engine has a built-in decompression system so your compression will be around 65-70 on good engine. Checking when lower will usually show up an out of spec condition. If it not this then an leak-down test is order to do a non-invasive cylinder condition test.
BTW the ultrasonic cleaner I have has been one the best investment I ever made; even though, I have up sized a couple times due working with larger carburetors. I even clean a couple fiber Echo trimmer air filters this week. When got through they looked new. What do you think about this before and after photo.