No thermostat on a 50hp 4-cyl.
Is the powerhead getting hot when you see indication of no water pumping? The exhaust plate should stay cool or lukewarm at most. The telltale could be intermittant due to debris lodged in the block, or issues with leaking exhaust plate/gaskets. In that case, the exhaust would blow into the cooling passages and blow out the water.
When underway, you should see a steady spray of water coming out the exhaust relief ports at the back of the engine, under the lower cowling. This and the exhaust plate temperature are better indications of water flow/cooling than the telltale.
If the impeller key were missing, it wouldn't pump water at any speed.
Maybe what you need to do is pull the lower unit and run water up the copper water supply tube. Find a rubber hose of size that'll slip snugly over the end of the copper tube, then flow water from your garden hose. You'll be able to see what kind of water flow you get out the telltale then.
And if you're extremely careful, you can fire up the engine, at idle speeds only, without the L/U installed. With a positive supply of water, there is no way exhaust can blow it out. Instead, you'll be accomplishing a pressure test of the exhaust system; any leaks will cause water to be sucked into the exhaust ports and you'll see droplets of water on the spark plugs, and probably notice erratic idle as well.
Be prepared to shut the thing down fast, 'cause without any load or backpressure it's gonna wanna rev like a Mad Dog if you give it any throttle at all!
HTH & let us know what you find...........ed
p.s. don't forget to replace that bad exhaust tube gasket, you don't want exhaust blowing around where it shouldn't.