Fun, I just went through this with a Bilresa remote.
IPv6 needs to be enabled on your HA server, I ran this command though I'm not entirely sure if that was required: ha docker options --enable-ipv6=true
But what really seemed to do the trick for me was, on my phone, going to Settings > Companion App > Troubleshooting (almost near the bottom) > Sync Thread Credentials.
Why there? Good question!
Why not the "Send thread credentials to Home Assistant" button at the actual Thread configuration? Equally good question!
This was with the Home Assistant Green and ZBT-2, mind you. I would've expected there to be some level of "out of the box" integration. Oh well.