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I suspect that the reason they have "their own special name" is less because of how prolific/important they are (though they are used heavily in some contexts, geometric optimization problems, etc) but more because their analysis and utility is typically very different from say R->R functions.
I'm not trying to understate their usefulness either though, functionals certainly are foundational to certain topics such as calculus of variations (as you are aware if you are looking at euler-lagrance) which is pretty well developed and has plenty of applications.
