I expect you are on the right track. In order, I'd check:
- Screen at transmission cooler (or similar) on intake hose. Debris in the screen that is forced through at speed, but not at idle? (Happened to me a couple years ago with wood chips - lot of sticks on the lake.)
The thermostat is possible, but I wouldn't think it'd matter whether at speed or at idle. Maybe at speed there's enough force to get more water through a narrow passageway.