I think one needs to consider the available customers in the Minehead area (and similar applies to much of Cornwall.)
The resident population is small and with over half of households suffering deprivation. (Not great)
Butlins as the main holiday provider only attracts a certain clientele.
It is distant from bus enthusiasts and any day trippers.
Many customers will prefer a reliable closed top bus to enjoy the views, than being blown away on a grey day.
The councils are effectively paying for a short holiday attraction, which does not run in the winter.
EX1 will work in a timetable search - Exmoor Coaster does not unless it is EXM. (The Falcon coach was unhelpfully FLN, but now FAL)
Holiday makers may well not remember route numbers from year to year and aren't necessarily on repeat visits anyway. They just enquire about the service - read the bus stop display, search online, etc.
When the Minehead outstation has completed its electrification works, then it can be used for what ever buses Taunton depot wants or can obtain, to give it.
If First pulls out of Cornwall, then Taunton could well transfer to Bristol, as they have already taken over 3 routes to Bridgwater, Taunton and Yeovil with Bridgwater depot closed and Yeovil transferred to an outstation on an industrial estate.