The result is even stronger: we can prove that it will sometimes not terminate!

Still that's not too bad, since type checking is always conservative: if a program passes, we know it's correct(ly typed). If it doesn't pass, we don't gain any knowledge: it may be incorrect, or it may be correct in a way which the type checker's limited algorithm cannot determine (ie. a Goedel sentence).

I'm very concerned whether my programs terminate/coterminate, since having to kill them part-way-through could cause corruption and other nastiness. Whether the type-checker terminates or not I don't really care about; I can just kill it after a certain timeout and keep fiddling with my code until it passes, just like any other type error.

Of course, a timeout removes Turing-completeness, but that timeout is under my control at the commandline, rather than being an inherent property of the algorithm.

This is only partially true. The type system is only unable to prove termination if you choose to use those constructs which may disallow termination.

I don't know about you but I'd rather not have my type checker hang and force me to interrupt the process all the time.