I wonder how many language designers explicitly thought about the goal of turing completeness or just had it happen as they added high level constructs like functions et al.

Certainly turing completeness does not guarantee a nice language (see brainfuck and friends) and I am sure that there are some non-turing complete DSLs which are very nice to work with. It may just be that when people try to design general purpose languages, they end up with turing completeness.