Below the steel housing of the t/stat itself is the aluminum body of the airvalve (extra air for cold running). The water leaks between these two. Beneath the aluminum housing is the pump body itself and the shim adjustments for these systems relative normal governing.
The only leaking I can think is happening would be coolant and I don't see what this has to do with your running unless it contributes to the airvalve sticking in its aluminum housing.
The t/stat itself is the only part serviced separately anymore (it is inside the top housing with the hoses connected), but like I say parts should be easily available used. I just don't see the point to solving the leak unless the pump can be made to perform correctly.
Bosch Master, ASE Master, L1
33 years MB technician