Been fighting this for weeks on my little handheld project and thought I'd dump what worked since the docs say it can't be done. The C6 radio does support the isochronous channels LE Audio needs, Espressif just never shipped the host stack for it. That gap is the whole problem.
The trick is splitting the job across two chips if you have them. I ran the stock C6 controller as-is and wrote a small CIS layer on top of it, then let the main chip handle the audio host side over HCI. If you're stuck with a single C6, it's still doable but the timing gets tighter, so start with the two-chip setup. Get a regular BLE connection to your earbuds first and confirm it before touching anything audio related. Biggest gotchas: the codec config has to match exactly what the earbuds advertise or you get silence with no error, and the reconnect logic needs to store the bond keys or you'll be re-pairing constantly. Also budget way more time for the sync timing than you think, that's where the dropouts came from for me. It's MIT licensed and up on GitHub, search esp32 le audio cis and you'll find it. If you get it running on other earbuds, post your results.