How to make setuid / suid shell scripts work the way setuid binaries work?

Issue

• What's the procedure to get root-owned setuid scripts to always be run with root permissions?

• When executing shell scripts that have the setuid bit (e.g., perms of rwsr-xr-x), the scripts run as the user that executes them, not as the user that owns them. This is contrary to how setuid is handled for binaries (e.g., /usr/bin/passwd), which run as the user that owns them, regardless of which user executes them.