I've tried various GTM click triggers in the past but have never gotten them to work, either. I believe we're running into issues because the click is registered in the
iframe the live chat widget creates.
Here's a similar topic I responded to recently.
As I mentioned there, I recommend upvoting and commenting on this HubSpot Community Idea and any other topics which match your use case to help them become realities.
This information is sent directly to our Product team and helps prioritize feature requests that have the greatest impact on our customers. I’ve already upvoted the idea and will also do my part and pass your feedback along through internal channels.