I'm pretty sure manual ED30s don't have the inline thermostat, well, mine doesn't at least. Not sure if the DSG might has one, but worth checking/replacing if yours is a DSG.
The OP's problem sounds like the main pump to me. The classic the faster you go, the hotter it gets. Coolant system pressure is about 1 bar, so that is some pressure for the impeller blade to overcome. What can happen with the plastic ones (as GVK said) is the impeller spins on it's shaft rather than moving the water. The mechanical bond breaks down over the years.
The mechanic was wrong to hold off on the belt/pump change as the pump is the heart of the coolant system, and the belt is long over due as well.
You might also want to check the secondary pump as well, which lives in the primary thermostat area. That pump supplements the mechanical one and also circulates the coolant after shutdown to prevent hot spots.
@exodus_ste , the fans can come on as part of that process, so I wouldn't be concerned about that. So long as they don't stay on permanently. The after run circulation process usually lasts for approx 5 mins.
As for the fans not coming on, if both of them work via an output test, it proves electrical continuity at least, so you need to investigate the signals from the sensors.