Every time I try to boot up the game it shows me an error saying "Sound System Not Initialized" and won't let me launch it, no clue as to what this even means. Anyone seen anything like that?

Im not. Old games often require audio devices. USB headsets is separate audio devices entirely with own audio chip (audio card) inside (at control panel with volume buttons in middle of usb wire). In nowadays Windows all headsets is separate devices.
I doubt there is issues with usb headsets. Vise verse - smartphone headseats are often cause issues. Or microphones.

Check Wundows audio devices settings. Which one default. There is two options for default (two checkbox). Cant say more specific for Win10+ (and each update build change OS settings entirely with no reasons). Make sure speakers are set (maybe)
Check input default devices which one set. DId you have build-in micro? Eg notebook/tablet PC.

Each device and plugged headphones have own settings in Windows. Experiment with audio quality settings keep in mind you need repeat that after you plug/unplug headsets or change default (Win7 store settings for each device, but now if you change it from main Windows Metro-style (mobile style) Control Panel i guess it work only for current default device).

There is (if current Win10 still have it) option for 'exclusive mode'. Try disable it (again, that checkbox will turn back when device changed)

Some overlay apps can interfer that "exclusive mode" mode. Try close (from background apps too) any apps. Voiceover ones first. Maybe even MSI Afterburner. They are differ for micro and nor interfer probably, but for speakers even Afterburner can cause issues. Overlayname.exe can use audio devices exclusively interfer gamename.exe.
Nahimic app is first one to close if you have it. Sometimes - especially since few last updates - evn Steam Overlay took audio stream. There s audio played from Steam browser tabs and silence in game.

When you get game working you can install fan-made wrapper-mod KainGL. This wrapper make game working always in windowed/borderless fullscreen windowed with few else features. Proper compatibility allow make true 60 fps. End subtitles support.
Windowed mode mean all video settings will took from Desktop. And can solve audio issues too. Maybe it can help you with game launching but that not the feature of it. Tefhnically game should work as is and wrapper is a mod it to better.

http://www.pcgamingwiki.com/wiki/Troubleshooting_guide/Audio_problems
http://www.pcgamingwiki.com/wiki/Blood_Omen:_Legacy_of_Kain

For compatibility GOG already use own GOG DX Wrapper (d3d and dinput/xinput) and also mod the exe file to slowdown the game. At least GOG version should work as is. No issues.

Maybe you need verify game installation if you use GOG Galaxy and Galaxy build. Or reinstall for GOG offline. This game did not use CDAudio. And have no audio compatibility issues (it can be broken along with whole game for Retail version doe to general incompatibility with too fast CPU/GPU).

So, I've made a few osts regarding GOG games not working and now all of them just work. I didn't do anything new with them, all I can think about is changing audio devices to the system default and that's it. I genuinely have no idea what solved all of those issues.