http://mathoverflow.net/a/153106
Because of the list of correct answers for a finite subset of the Halting Problem is finite, the Turing machine you mention does exist, but we not only can't find it, we can't know when we've found it!
Where the Math Overflow post mentions that "experts could compute the particular value of n", there has recently been such a bound published in a thesis, such that we can be confident that we can never construct or recognize the solutions beyond that point (using a particular axiomatization of mathematics).
You referred to the idea that "you can have a Turing machine that solves this", and we can agree on that with the caveat that, above certain problem instance sizes, you can't know or verify in any way that you have such a Turing machine!
My argument stands. Unless the halting problem is overcome, there will still be jobs for humans to write Turing-complete programs.
Turing machine X can produce programs that meets certain specifications => Turing machine X solves the halting problem.
There is a fairly large subset of useful programs that can be proven to halt. Anything that uses straight-line linear control flow. So can anything with that plus conditionals. So can that plus foreach loops, as long as iterators do not reflect updates to their underlying collections. Add a "forever { ... }" construct and you can prove that the program will not halt; in combination with the other constructs, you can prove liveness on each request handler while also guaranteeing that the server itself will never go down.
The two constructs you have to watch out for are loops that mutate state used in the conditional and unbounded recursion. Even for these, there are techniques to increase the set of programs that can be reasoned about, eg. using dataflow analysis to identify which state is mutable and preventing it from being used in conditionals or tracking data & codata through the typesystem.
http://blog.sigfpe.com/2007/07/data-and-codata.html
Such a language would not be Turing-complete; you won't be able to write an interpreter for a Turing-complete programming language in it. But the majority of common business problems don't require an interpreter for another programming language; most of them focus on storing data, triggering events, or computing functions of data.