What chroot() is really for

Non-executable /tmp can be trivially circumvented by calling the loader with your binary as an argument.

Then you'd probably want to make sure that loader isn't present in your chroot jail. I assume you're talking about glibc's ld-linux.so, which is an essential part of running programs that use dynamically linked libraries. Chroot jails I've seen have statically linked programs and don't need it.

Incidentally, if this is really an issue -- people want to have shared libraries within a chroot jail and still stop people from running programs they created themselves in /tmp -- it would not be hard to disable ld-linux.so's ability to run programs like that. The ability to exec ld-linux.so is a frill added to its basic function as a Linux program interpreter that runs when you exec something else.

That was true some years ago. Nowadays ld-linux.so fails if it is called
on a binary on a non-executable mount.