A lot of distros that don't use systemd still use sysvinit. That one works just fine but it can only start services concurrently so if you have a lot of services you aren't getting an ideal boot speed. It isn't 'next-gen' and therefore not quite a "replacement" for systemd but it works just fine. What qualifies better as a replacement is runit or s6. But older things like openrc and upstart are also just fine. However, systemd is so integrated that you cannot freely change your init system, even though that was quite easy in the past. Now you need a dedicated distro for it. Some good systemd-free distros include Devuan, antix, Artix, Slackware, Gentoo, MX Linux, PCLinuxOS.

For some reason NOPASSWD on a single command didn't change anything for me on Slackware, I was only ever able to get it to work with NOPASSWD:ALL which is too permissive for my taste these days. I'll try this line however, maybe it will work with /sbin/shutdown (which is where that program is located on Slackware).