Hi @timjcraig,
Yes it should be possible to select private groups you are a member of, I think we had a bug in v3.0 release where it was accidentally removed (ticket here tracking the issue).
You may still be able to override the channel to choose a private group by including "channel": "channel-identifier" in the JSON payload, where the channel identifier is the end portion of the URL of the private group you want the webhook to post to.