Hard to advise for sure.   Is this engine new to you?  Does it show signs of overheat damage in the past?  Fresh or salt water usage?  Does the engine show signs of previous work done (excess sealer/new gaskets)?  The exhaust cover gaskets are usually not an issue for these engines, unless debris from a failed impeller are blocking water passages intermittently.   If the engine has seen alot of salt water use, then aluminum eroding away/splitting off the cylinder sleeves is more of a concern.

I guess I would start at the beginning.  Are you sure the water pump is not an issue?  Are you using OEM or aftermarket parts?  Are you sure the threads on the water pump screws/holes are in good shape and clamping properly?  Perhaps the tiny vent in the impeller housing is plugged up.  Perhaps the aluminum under the SS impeller plate is worn/distorted/eroded creating an air leak into the water pump.  Is the water tube in good shape, fitting properly into the impeller housing grommet?  Oftentimes the water tubes get distorted/bent when misassembled.  Once bent, they no longer fit securely/properly in the impeller housing water tube grommet.  You could remove the impeller housing, slide it up inside the exhaust housing into the water tube, the bottom of the impeller housing should be pretty much “flush/even” with the mating surface of the exhaust housing.     I guess the simplest thing to do here would be to try another “known good” gearcase/water pump assembly if you have one.

I have seen the intake screen behind the prop be full of sediment. This is very hard to see. Then the motor only gets water through the tiny holes in the bypass cover, which will be ok at idle for a few minutes. Something to think about.

most  of the water does not even travel via the thermostat so it should  always  pee.

if it stops after getting  to running temp the water pump would not know anyway so I suspect a powerhead area  problem and  not getting anywhere I would pull the head and inspect for  something blocking the  water passages. run  air pressure backwards  from pee hole and thermostat water inlet  . pull the exhaust cover to inspect

just in case….the water flow diagram  shows  both the cold and  “cyl  warmed up”  water exiting the thermostat  mixing together when the running temp is reached .  the  mix  gets  dumped out

the common area for both cold and hot being   the thermostat cavity  and  if BOTH cold and hot water suddenly stop flowing out and down ……..check to make certain the “thermostats  area”  isn’t causing the problem..even if you pulled the thermostat to test .

try to run  some mono line  in all  water passages you can reach