The thermostat does not need to be open for water to flow out indicator, it's simply an indication of water pressure, if the thermostat were stuck shut it should still "pee."
Those intakes definitely look like they are submerged at all times. You say other boat drivers have cost you impellers, what do they look like when they failed? Is the "hub" spinning as Joeer77 questioned or are the pump vanes damaged from lack of water in the housing?
I have burnt through several as at first, I thought it was just a failed impeller. I would replace it test run the motor, and think everything is good. I would then take it out and travel at high speed and it would fail. Many times I thought my work was the problem as it failed for an unknown reason. It was hard to diagnose as the only time there is a problem is when traveling very fast. When traveling at 20 MPS in a 10-foot dinghy, the last place you are looking is around the motor at the water outlet as you don't want to crash into anything. So a few water pumps failed while I was out the helm. It was about my 3rd on where I went out and had a friend driver while I laid back on the transom.
When I pull out the water pump it is mostly suffering from vanes that have the tips melted off and melted rubber filling the outflow port on the water pump housing. I have worked diligently to recover all the pieces and replace any damaged parts.
I own about 8 outboards and have done about 30-40 pumps in the last several years and feel my skills are up to doing a proper job. When idling and at most speeds the pump works well. When coming out of the "hole" the high RPM engine and low speed boat pump water just fine. Only when really going fast do I suffer failure.
I am thinking about positioning some gopro under water shooting at high frame rate and running over them to try to find the problem. I am still hopeful someone here might have and idea and I appreciate all suggestions. Thanks!
