You eliminated quite a bit of stuff.
What I would think about doing is check the valve lift. compare the exhaust push rod left to the intake. Compare the lift of each to each other. Look for as much as 1/4 inch difference in lift height, I've seen one lobe loose enough lift on those 28 engines, especially the intake and the compression release still operate but not get up to speed and sometimes detect by ear a sputter through the carb throat.
(why, I've seen a lobe wear down to where lift would not be correct)
You can find exact specs for the lift in a search.
You say you have adjusted both valves to around .005 but the lift on the intake looks suspicious.
If you find the valve lift ok check the flywheel key.
In your first post it appears as though you have never seen this motor run good. (you got it as a used motor to swap onto a mower)
Have you checked the magneto air gap as being .010-,012 and swapped the ignition magneto and also checked with the proper spark plug?
I've also seen ignition modules do strange things.
Do not operate that engine over about 15 seconds with the air cowling off, they will overheat fast.