Sorry for necroing; hopefully it's excusable in this instance. Just wanted to post here again to offer closure.
Got discouraged and didn't touch Civ3 for over a year and other stuff took up my time. Had that itch again tonight, though, so I finally circled back and triple checked everything.
In short: I finally got it to work!
Turns out this was a lie:
Done the registry changes both under HKEY_CURRENT_USER and HKEY_LOCAL_MACHINE
I did another pass of my registry and noticed I somehow completely forgot (and didn't notice that I forgot) to add the CLSID entries under here:
Computer\HKEY_CURRENT_USER\Software\Classes\WOW6432Node\CLSID
Made the changes necessary and I can finally hear the music again. Oops.
Triple check that you've followed the instructions, y'all. I feel awfully silly now
Thanks lethal & Lord_Hill for trying to offer assistance.
(2024-11-30) Edit: Also noticed that running the game elevated (i.e., as an administrator) also breaks sound for me. It's possible I messed something else up on my end that makes it break in this manner, but not running the game elevated makes it work again, so maybe something to think about for anyone who might be reading this in the future.