I agree with your point about functions versus algorithms (or computation if you prefer that term), but I disagree that your definition of functions isn't useful for computing.
It's precisely that sense of function that things like Haskell (or Erlang, or Prolog) try to introduce - a different syntax for writing the same value.
A pure function is just explaining that a particular value is functions applied to other particular (stand in) values (and a way to introduce names to values). These are only useful for explaining which computation you'd like to happen when the program actually goes to carry one out - I'd like the number which is the same as the cosine function applied to 5, which tells it precisely how to compute that number (assuming it can build an algorithm equivalent to cosine). Of course, we could use another pure definition (that cosine of x is the same as a series expansion about x) to allow the compiler to replace the desired cosine function with something that has intrinsics on the system - evaluation a polynomial at the point 5.
So in essence, we can use functions as a tool to describe what value we want from the program (or various structural properties), in such a way that the compiler can correctly infer how to build an algorithm to do so, and tie it correctly in to our (actually) algorithmic code.
Minor aside: functions (in the math world) apply in contexts besides the real numbers, including being syntactic notation for binary numbers (or fields of order 2^n).