it is easy to visualise it being cool
My gague, and your comment indicate plenty water.
Our old Volvo exhaust water was bathtub temperature, say 120 F, as judged when we used it to heat up a freon foghorn in cold weather.
The thermostat regulated the circulating cooling water ( fresh side) so can keep the engine at the right temperature even when ther is "too much" seawater flow to the exhaust