Yes, this is absolutely correct. If you give up on Turing-completeness you can prove that a subset of programs halt. A computer could search this infinite space for programs that solve a particular problem. However, there may not be a program P in this space that solves the problem in question so the problem of finding P in the subset of provably haltable programs does not necessarily halt.

My argument stands. Unless the halting problem is overcome, there will still be jobs for humans to write Turing-complete programs.