The little disk will rise when temp is reached.  You probably didn’t get it hot enough.  When they fail, they almost always fail in the open position.  Almost.

It doesn’t actually “switch to internal cooling”.  There are two water flow paths through the powerhead.  One path is always open to free flow and is not thermostat controlled.  The other path is thermostat controlled and flows according to thermostat state, which is constantly sampling and opens/closes as necessary.  Danged if I can tell the ‘stat position by observing the discharge.

I never knew there was rubber in them! Thanks for that great diagram! Very funny about the one on eBay. Although a little troubling….

Like Frank says, thermostat operation would be near impossible to judge from the cooling water outlet on this engine. I would be more suspect of a worn impeller/housing if the outlet water volume seems to drop after start up. Grooved/pitted impeller housings are often overlooked as a source of poor water pump performance.
Have you had the gearcase off lately? I only ask because these engines are low in the water and more prone to spline seizure, especially in salt water. So, be very careful if you are removing the gearcase for the first time. Forcing the gearcase down/off, while leaving the driveshaft stuck in the crank splines creates quite a mess.