You need to pause and think about how that system works. It is a thermostat-controlled recirculating cooling system. When you start the motor, the water goes up and circulates through the powerhead. Then it is presented to the thermostat. If it is not warm enough, the water is directed through that plastic check valve and goes back down to the water pump for recirculation. Back up for a repeat of the first trip, and if still not warm enough, back to the water pump. This can go on all day if the lake water is cool and the motor is not developing a lot of heat (slow speed, and a good water pump).
Now, even while in recirculating mode, a bit of water is discharged. This is necessary to cool the exhaust (and that electric shift cable). Also, a bit will spritz out the above water exhaust relief as a tell tale. Because some is being discharged at all times, new lake water has to be added to make up for it. THAT is why it can run so long with the 'stat remaining closed.
When you run at higher speeds and/or loads, the powerhead will get warm enough to open the 'stat. That is what happened when you briefly saw a heavier flow, normal for a warmed-up motor. BUT. remember, it is not a thing of the 'stat being either open or closed. It is constantly sampling the water temp and opening and closing as necessary to maintain the proper temp.
I typed all this simply to tell you that there is nothing wrong with your motor. As long as it is not overheating, as you indicated, it is acting normally, as designed. So don't "fix" it.