Wireless controller doesn't work with wired headset, but still passes sound tests

Perhaps someone can help me with an inexplicable audio problem I've been having. Today, I went and bought a new wireless Xbox One controller; from Microsoft, not aftermarket or anything. I then started a game to test the controller with a wired headset, which I know works. No sound. Here is a list of things I subsequently attempted:

1. Controller synced? Yes. No sound.
2. Headset jack fully seated into controller? Yes. No sound.
3. Headset muted? No. No sound.
4. Audio all the way down? No. No sound.
5. Cleaned headset jack with rubbing alcohol. No sound.
6. Went to Xbox One accessories to see if the headset was being detected. It was.
7. Updated the controller firmware, with the headset plugged in. No sound.
8. Hard-reset the Xbox. No sound.
9. Went to Xbox One settings, turned headset volume to max and mic monitoring to near max. No sound.
10. Checked other audio settings (uncompressed, Windows Sonic for headphones, etc.). No sound.
11. Downloaded Skype onto my Xbox in order to test the sound and the mic. Both of them worked?! Multiple tests confirmed they work.
12. Tested headset with different controller (actually on 2). It works on a wired controller I have, but not on another wireless controller (though this one may be broken, because headset worked on it in the past).
13. Tested controller with different headset. Mixed results; one time I get no sound, another time I get sound.
14. Checked Xbox One privacy settings. Communication both ways is open to everybody, so no fix here.
15. Checked controller battery level. They're brand new, and it's fully charged, so no fix here.
16. Started a party to see if my mic was being detected and my little profile icon lit up. It did; still no sound.
17. Another hard reset. No sound.
18. Extensively searched online for this same problem. No success.
19. Briefly searched on this forum for this same problem before posting this. Nothing.

After all this, we managed to find the USB transmitter to a wireless headset; I plugged it into the Xbox, turned the headset on, and it worked. I then got another player to confirm that I can hear them, and they could hear me, through this wireless headset. So what could be the problem? Both the controller and my wired headset, individually, work. They even worked together, when I did the sound tests through the Skype app on my Xbox. I haven’t tested any other app like Youtube or anything.

Has anyone else had this type of problem, or is there anyone who thinks they can help? I would appreciate it. (PS mods: this is my first forum post here. If this is in the wrong category of posts, feel free to move as needed.)

:)