Why I can't set the keys on my remote device to change my Mac's volume?

Check the function key settings in System Preferences > Keyboard.

There is an option there to "Use F1, F2, etc. keys as standard function keys". If that is check marked, then you will have to include the press of the fn key along with the F11 and F12 keys to decrease/increase the volume. Those would be the key combinations to assign to your devices. (fn-F11) (fn-F12)

If that option is not check marked, then the function keys alone should decrease or increase the volume and the key assignments would simply be F11 or F12.

Keep in mind that these devices are nothing like standard keyboards, nor is the software that drives them. The onus of keeping the software updated and compatible with macOS will be on the device manufacturers. They should be your first resource for information about setup and configuration.

I am well aware of the access Apple affords developers.

Since you are unable to assign these simple keystrokes to either of your devices, I suggest that you have a mapping conflict elsewhere in your system. You should examine all of your various key assignments closely in System Preferences.

☮️

What is the remote device?

Might try Karabiner...

https://pqrs.org/osx/karabiner/ 

You can also easily make custom keyboards with...

http://scripts.sil.org/ukelele

Thanks to Old Toad…

You can do what you want with Keyboard Maestro and a whole lot more .