Pool Company Telling me to connect Motor to 240 Volts for Longevity

Right, of course. I suppose since the load is in constant flux (as thousands of people turn things on and off), so is the voltage. I'm now picturing a giant regulator of sorts, somewhere in the mix!

So when manufacturers provide spec's for their products, they just pick one of those voltages and they're common enough that everyone "knows what it means," and it is implied that the product will work without issue within some variable range, I guess. Thanks James.
The giant regulator is actually the governors of the steam turbine generators at the power plant maintaining a constant rotor speed. As load increases, generator rotor starts slowing down due to increasing flux fields, which causes governor to admit more steam to generator turbine to maintain rotor rpms. In a sense you are correct because as more power is needed, more steam is used, steam boiler pressure starts to drop, and a fuel pressure regulator opens wider to maintain boiler pressure.
 
As an added point, the motor was indeed running very hot this summer. Is that a normal symptom of a "dying" motor or does it seem likely that something was off in the rest of the pool system?
Probably a dying motor. You can check the voltage and current when the new motor is running to make sure that they are correct.

The pump is likely oversized for your application. Maybe consider going to a smaller motor and impeller to reduce the power needed. How much flow do you need?
 
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