OK, so plenty of water coming up to the thermostat. The cool red dye shows that the passage out of the block is at least somewhat clear…
Does the engine actually overheat?
Any evidence of water on the plugs? Just trying to think of why the water would exit when not running, but not when running… And only one thing comes to mind.
Perhaps the head gasket is blown in such a way to block water flow. Perhaps the water is getting so hot it is boiling off before reaching the relief port. Any signs of hot steam coming for the exhaust relief?
The water exits the block through a very sharp elbow at the bottom of the block on the stbd side. Salt and debris gets caught in here easily. You mention that the engine is a fresh water engine, so salt build up seems unlikely, although sometimes even fresh water leaves a crusty debris, along with those crazy mussels.
Perhaps the base gasket is messed up, or there is a defect in the exhaust hsg cavity allowing the water to “short circuit” into the exhaust housing instead of flowing out of the relief port.
So, I guess I would start with a simple compression test. An uneven reading could indicate a bad head gasket. If compression seems OK/even, and the engine actually overheats, then I would probably pull the head and powerhead. The bolts shouldn’t put up much of a fight because this is a fresh water engine. Have a real close look at that tight water elbow on the bottom of the block, look for crud on the cylinder water passages, and check those thermostat passages carefully as well. Maybe we should start with the thermostat passages. If one of the passages is plugged, it may prevent water from exiting the block… But, now I am remembering you saying that the engine behaves the same way even with the thermostat removed, so this theory seems unlikely, although it might just be one of the channels/passages in the thermostat cap has come crap in it preventing the water from exiting. Start with this theory before digging into the other stuff, even though I didn’t think of it until the end of my reply. Perhaps someone used the wrong thermostat gasket, or made their own as well. Always start with the simple stuff!
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