This needs moderating up. Talk to an Ocaml programmer and a Haskell programmer about what makes a functional language and you'll see very different opinions and these two are languages that were actually designed as functional languages: the bits that end up in other languages are a tiny subset.
Coming from the Haskell side, I see functional programming as programming without side effects and with monads. You can implement monadic constructs in other languages, but it rarely makes code cleaner. Just having higher-order functions doesn't make a language a functional language any more than having structs makes C an object-oriented language.
If the question is 'do you think using higher-order functions simplifies the expression of some algorithms' then the answer is obviously 'yes': programmers have a lot of tools to choose from and most of them are useful at least some of the time.