Hi, I have now also this issue in my cleanly installation 3.8.50 from zero with OpenRC..
Replying my own comments but this one is for @Thanatermesis .
'elive-pm shutdown|poweroff' points to 'sbin/poweroff' which is, on a systemd machine, linked to 'systemctl' but -..... on my openRC machine it points to '/usr/sbin/halt', which does not completely turn off the machine but merely halts the system. It leaves the power on and will eventually drain the battery. 
It would be better to use '/usr/sbin/poweroff' .... that does power-off the machine (laptop in my case) and doesn't require entering a sudo password either. 
I did notice that 'sudo reboot' works from a terminal but NOT from the e16 menu option nor from E27 menu..
On top 'reboot' links to 'halt' but actually does reboot the ,machine. I have no idea why that works.
Addendum
Scouring the web I discovered that 'lightdm' uses 'consolekit' whereas 'lxdm' uses 'elogind' for authentification. 
And ..... 'entrance' doesn't show up at all on the openrc system, so that's not an option. 
Addendum2 (sorry, learning as I go along 
)
For 'lightdm' to, at least have the shutdown|reboot options available there's a small edit to be made in '/etc/pam.d/lightdm-greeter' where there is a line:
session optional pam_systemd.so
that has to be changed to:
session optional pam_elogind.so
This will NOT repair the shutdown|reboot buttons in the desktop-menu but will at least give a GUI method in lightdm when logged out using those buttons.
Aah, good one, ADDED.
But, on the other hand, what is very important is to make the gksu alternative to work on openrc installs, can you make a look to /bin/gksu ? it is basically a script wrapper since gksu doesn't exist since bookworm, so we can make the wrapper to use an alternative when being in openrc... I already did that but seems like it didn't worked ? you may add "set -x" into it to see which command try's to use
ok that's another thing... so elive-pm should point to "poweroff" for the non-systemd cases instead of halt, right ? what is the correct / full command for that that works for you ?
We should better try all that in trixie (not yet), and/or we can patch it. About entrance is not yet clear if we would use it or just make a "greeter" for lightdm using and EDJE/EFL interface (because lightdm core is much more featured, allows multiple users running at ht esame time for example, among other things especially compatible)
Adding the line (instead of replacing) will also work ?
Maybe running a massive sed -i to the directory files when moved to openrc:
/e/pam.d ❯❯❯ ack pam_systemd
common-session
23-session required pam_unix.so
24:session optional pam_systemd.so
25-session optional pam_ecryptfs.so unwrap
lightdm-greeter
17-session required pam_unix.so
18:session optional pam_systemd.so
So... we have multiple bugs / things to improve here, on this order, for openrc installs:
gksu should work (it is used by many applications, not only shutdown)
elive-pm should work on openrc installs
lightdm should shutdown & reboot
e16 & e26+ should work too
Will do, given time and opportunity.
Simply "sudo poweroff"
Not 'only' for non-systemd IMO but simply do not use "halt"
I'm actually even not fully in the clear why "halt" is used. Maybe for desktop machines and remote booting ??
The difference is the power-unit being turned off or not. I suspect it's an acpi call.
Entrance isn'teven stable yet IMO. So 'lightdm' is the better option and on top the one most used by the 'openrc' devs.
Albeit, I found 'lxdm' quite nice too ...... clean and simple, doing it's job well. 