A failed water pump nearly always makes a lot of noise, spews fluid or siezes up tighter than a tick. VERY rarely does the impeller fail and usually only gets less efficient through either poor design (some aftermarket stuff is Caca) or cavitation "chipping" little bits off the impeller or casing.

Thermostat and head operate together to provide a bypass system when the thermostat is closed. Anyway it would not cause a loss of coolant.

Blown head gasket losing half your radiator water would be very apparent (plumes of steam, hydrolocked engine, bubbles and goo in radiator or mysterious goo in sump).

My suspicion is a bad hose connection or hidden leak in the radiator or heater core. Having these serviced (if you pull them out) is cheap insurance, even if the problem turns out to be elsewhere.

You just may turn up another issue when the radiator comes out.

As a side note, I do not compeltely trust any of my gauges. If the engine does not act hot or feel hot even when the gauge says it is hot, I would get a thermometer in the radiator head tank with the cap off to check the thermostats veracity.
My Dodge with the Cummins (1997) has a spastic thermostat that likes to periodically wave at me between 170 and 240 degrees F. Very unnerving but wildly inaccurate... makes me stop and verify myself periodically, however.

Mike!