In fact I'm almost positive that other cross toolchains I've worked with in the past have done just that.
But — I can't seem to be able to find the right combination of incantations. Between all the --prefix, DESTDIR, --with-sysroot, --with-byild-sysroot, --with-headers, --with-native-system-header-dir, something goes wrong. I was actually able to convince GCC to look for headers in the right place, but the libraries still escape me.