It's got nothing to do with what the note's called, and nothing to do with with whether A=440 or any other number.

But there are people who say they would be uncomfortable with the pitch being wrong if for example they heard a record playing a touch fast, eg. A from 440 to 442.

Whether they are aware of it or not, the note played by the musician would be A.

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But I guess it could be useful to have an internal tuning fork ...

Maybe but playing in pub sessions I have to be in with others who may not be in A=440 (or whatever my tuning fork is set to) and may not even all be tuned to the same A or perhaps in some cases not use the same tuning system - it can be a case of trying to find a sort of best fit rather than taking tuner out, establishing A=442 and expecting things to work out on the tuner from there.

If (as it seems there may well be) it resulted in a conflict, I'd rather have my insensitivity to these things.