I have the sticky bit set and the file is owned by root:

Just as a terminology clarification, this is the setuid bit, not the sticky bit. The sticky bit has a very different meaning from setuid bit or setgid bit. It also behaves differently based on if the target is a file or a directory.

The documentation and behavior of what the sticky bit does is typically not well understood among even experienced system administrators (my own experience), partly due to poor, incomplete, or inconsistent documentation of the bit and infrequent use.