> I get that being able to check for non-termination would be useful—but is there more?

It's not just that "there is more", it's that there is everything.

The halting problem which we're talking about here is actually easily generalizable into a set of generic theorems about static analysis of programs in general. The usual first step is to prove the first incompleteness theorem (no axiom system proves all of the interesting facts about the integers), then modify that into a proof of the second incompleteness theorem (that every logic which proves its own consistency must be inconsistent). But it goes way further than that to Rice's theorem: every nontrivial semantic property of a computer program is undecidable. Put another way, the only way for the compiler to know that something is true is for there to be a really simple up-front convention which makes it true.

Suppose you want to make sure, statically, that you never dereference a null pointer. This says that the only way a compiler could ever be strong enough to prove this is if it's syntactically impossible to create a null pointer in the language in the first place, so that it's a "trivial property" of all programs that they never dereference null pointers.

Suppose you want to prove that a certain block of code is the fastest possible code which does a certain thing. The only ways to do this require the language to "normalize" all of its expressions so that there is only one block of code for everything you want to do. If there's more than one way to do it, then the analysis "is this the fastest code?" is nontrivial and you can't prove it! (You can always get heuristics of course to try to cover the most common cases, but your heuristics will never cover everything you want.)

Suppose Microsoft wants to fix Word and Excel's macro systems to a new form that cannot contain viruses: the only way to do that is if the macro system is either (a) Turing-incomplete, so that you can analyze its programs, or else (b) unable to do the virusy things that you're worried about in the first place. Otherwise there will always be a macro-virus which passes the virus-checker.