The default daily quota limit is inadequate to properly use the https://www.googleapis.com/youtube/v3/liveChat/messages endpoint. A query that returns you the necessary information costs about 8 units, and you get 10,000 units per day. Even if you're only querying for new messages every few seconds, that's barely enough for an hour.
I tried to increase the quota limit, but it appears that the form to do this doesn't work:
https://support.google.com/youtube/contact/yt_api_form?hl=en
It simply gives me "There were problems sending your form. Please try again."