I just recently brought an A4 back from the dead. One of the issues I had when it got running was overheating within a few minutes of starting.
Tested the water pressure coming out of the water pump.
good.
tested the water coming out of the thermostat cover.
good.
tested the water going into the exhaust manifold.
good.
tested the water coming OUT of the exhaust manifold.
very little and erratic flow.
Removed the outlet water line from the exhaust manifold.
Stuck a screwdriver in the manifold water outlet and swished it around.
Big clod of black goo came out.
Tested water flow out of manifold outlet.
Water gushed out.
Put it back together, has barely gotten warm since then.
Can be simple.
Start at the pump, work your way along the water flow, till you find something that doesn't flow.
Its also important to know whether you have a fresh water cooling system or a seawater cooling system.
Seawater pumps water thru the motor, then out the boat.
Freshwater goes thru a water cooled radiator.
What kind do you think have?
You said there was nothing under the thermostat cover.
That means there is nothing in the motor to slow down the water flow.
but for heat to happen, the water flow has to be blocked someway.
In my case, blockage of the outlet exhaust manifold was the cause.
The valve you speak of is just a method to get water to either go into the motor for full cooling or have some of the water bypass
the motor, so it warms up.
Valve closed = no bypass water, all runs thru the motor. Lots of cooling.
Valve open=some water does not cool the motor and just leaves. More warm.
This is the bypass valve on my engine.