to execute man-made programs and produce hopefully useful output
The ubiquity of computers is propagated by the mediocre to occupy harmlessly those who would otherwise bend their intelligence to taking over the world. Most of the money-laden power-wielding people you meet boast of not being able to use a computer - there is a lesson in there somewhere.
The purpose of computers is to imitate other machines. That's why Alan Turing dreamed them up. The fact that they can imitate more expensive machines makes them valuable. Imagine the cost of a mechanical equivalent of Excel.
You're forgetting Charles Babbage.
No I'm not. Charles Babbage didn't dream up a machine that could imitate any other machine.
Not clear - writings about the Analytical Engine suggest both he and Ada Lovelace saw the possibility. Alan Turing certainly made it explicit, though.
To a large extent, the purpose for which mankind uses computers depends upon their capabilities at any given time. Computers excel at:
information search and retrieval
They are fairly good at:
sound recording and playback
They could be used for:
human creativity enhancement
To occupy the masses when they get bored of lousy 'reality' television, appeasing them with video games, LOLcats, rage comics, and inane social networking posts so that they don't get restless enough to try to change the status quo.
See also, Software Is Really Pointless, The Purpose Of Machines
See also, The Milking of Humans