Yeah, but that's the 'cumbersome' part. Could we imagine a non-TC language with simple syntax that's suitable for at least some of the task you'd use a TC language for?
C compiled down to eBPF (as implemented with the Linux kernel) is not turing-complete. eBPF has a wide variety of uses from network filtering to hooking into certain syscalls.
The bytecode is theoretically turing-complete, but the verifier present in the kernel ensures that the code contains no loops and accesses no out-of-bounds memory.
With `clang`, you can compile C down to eBPF bytecode. It's a subset of C, with a lot of restrictions, but it is familiar and capable of some remarkable things past just simple packet parsing & filtering.
A useful subset of SQL is not turing-complete and is suitable for meaningful computational tasks. Datalog is also not turing-complete and is similarly useful.
Sure, more examples are CSP solvers and AMPL. They are all somewhat specific though. I was wondering if it would be possible to make one that's "general purpose" in the same sense you call programming languages "general purpose", i.e. one you'd write application software in.