I don't know much history, but am inclined to disagree with most of your claims (your statement about group theory is completely correct. I might be able to salvage my claim by restricting to the subset of group theory I care about, which is really more linear algebra and representation theory, but I don't know if the history would support me even then. Apologies for my error)

It is unlikely that the true extent of the "practical" importance of calculus today could have been predicted by even the most imaginative of Newton's fellow Cambridge dons in the late 17th century.

I don't understand this. Newton made the observation that calculus described not only the orbits of the bodies but also the behavior of the everyday objects humans interact with (and in fact described motion in general) before formally developing calculus---at least, thats how the normal version of the history goes (I have no idea how accurate it is). Are you claiming that an intelligent observer would doubt the importance of describing the motion of objects around them, or what?

The actual "physical" roots of the subject were more indirect, via differential/integral equations and the calculus of variations (18th-century physics, in other words), and it was basically the result of mathematicians' attempt to turn these somewhat ad-hoc disciplines into nice-looking abstract theories.

I was talking about the applications of functional analysis to understanding differential equations, which are (as I understand it) the actual point of functional analysis. Not coincidentally, functional analysis was developed in response to the obviously important problem of understand differential equations. Its not like someone sat down and developed functional analysis, and then it happened to later be discovered that it was a powerful technique for understanding the world.

Happening to provide a formalization for quantum mechanics is really not important in my view. If you think that no formalization of quantum mechanics would exist if mathematicians hadn't thought of functional calculus, I think you are very confused.

If you don't feel satisfied doing math, or think you could make a better contribution doing something else, you shouldn't be doing it. But don't make the mistake of pretending that your argument generalizes.

I am actually curious knowing whether I should be doing math. If I should, my life is much easier. I would like to have an honest discussion about the utility of math apart from specific applications. I tend to agree that the use of math is not in immediate applications. I also believe that you can foresee that calculus is useful, or differential equations, or any of the other things I mentioned, or even negative or imaginary numbers, and that this is not just hindsight bias but actual discrimination. This seems like a factual question which we have some hope of resolving (though not too much).