Whether 0.031" is enough depends on two things Paul: the rocker arm ratio, which for most pushrod engines is usually somewhere around 1.4, and the compliance of Tecumseh's springy-looking decompressor mechanism. Noting that you had to lift the valve itself by 0.012" less decompressor compliance to get adequate decompression, it is obvious that 1.4 times 0.031", or 0.043", is several times as much actual lift as you need for the job. However in practice you have found that the decompressor is not working. It therefore seems the decompressor mechanism's compliance is using up nearly all of your lift provided by the pin. Can you take that dimension across the camshaft and pin in two ways: with the pin being pushed down hard, and with the pivot arm and shaft on the opposite side of the cam being pushed up hard? If you are able to do that, we'll know the total compliance of the mechanism, including the slack. I'm guessing that most of it will be due to slack between the pivoted cam, and the tiny shaft it pivots on. I suspect that most of that slack will be due to wear.
With regard to the condition of the bottom of the exhaust tappet, I don't know how it got that way, but I think the decompressor pin only touches it way out near its periphery, not in there where the mark is. To see whether it is having any effect, you could check that the inlet and exhaust tappets are identical - they usually are - and try interchanging them.
The thrust washer is necessary, both to keep from wearing out the wrong parts, and to locate the camshaft axially. I'm concerned as to how it came to be missing: has that engine ever been dismantled before?
