[segfaultbuserr]

> What is special about the 'wheel' group and what is su even "checking" in the first place?

By convention, "wheel" is a special Unix user group that determines who can use "su" and "sudo". Most "su" and "sudo" implementations allow the sysadmin to make their use exclusive to the trusted users inside the "wheel" group. In most systems, it's the default setting of "su", and optional for "sudo" (given as an example in /etc/sudoers).

> if someone has the root password, can't they change what groups they're a member of?

No. If "su" is configured to be "wheel"-exclusive, you can't log in as root even if you have the password, because you cannot use "su" - unless you have direct access to the system console that allows you to type "username: root", which is almost never the case on servers that disable remote root login.