Operating Temperature Range question for Inrush Current Limiting Thermistors
The failure fairly likely is not related to the charred part.

Inrush thermistors could get very hot and having one charg glue woul not be a surprise BUT would probably be poor design of care-less manufacturing.

Having glue over a thermistor MAY reduce its heat dissipation and lead it to get even hotter than it was designed to - but having one exceed its max rated 
temperature would be no great surprise.

If you know what the thermistor is you may have a data sheet. But even without one -


Measure thermistor resistance in still air.
Place thermistor in hot air flow from a fan heater.
Resistance should change markedly.
Remove from hot air flow - resistance should change back exponentially towards original value
When the thermistor is at room temperature, clenching it in your hand while measuring its resistance will probably produce a noticeable change in resistance. 


I had a TV that behaved much as described. This was unrelated to an inrush thermistor. Some power supply capacitors went low capacitance with age. Regardless of why, replacing them fixed the problem. There is no good reason to think this is/was the problem with the set concerned - just a parallel example.