It makes sense to drop Turing-completeness. Many interesting languages are not Turing complete.
Perhaps the most practical example: just-in-time compilation of regular expressions to native code via LLVM. Regular expressions are not Turing complete.
Another example: take Agda as a source language. Agda is deliberately not Turing-complete, but can express almost any program you might be interested in. (Basically, Agda only allows you to express terminating programs. That's why it's not Turing complete.)
Perhaps the most practical example: just-in-time compilation of regular expressions to native code via LLVM. Regular expressions are not Turing complete.
Another example: take Agda as a source language. Agda is deliberately not Turing-complete, but can express almost any program you might be interested in. (Basically, Agda only allows you to express terminating programs. That's why it's not Turing complete.)