Hitting the pause button while watching a streaming live even on air.mozilla.org, doesn't pause the video at the current position, it just stops video from playing at all. Hitting play again doesn't start again from from where you left off, it restarts from the current live position. This is contrary to use expectations. If I want to step away from the stream for a moment, I want to pause and resume, not mute and refresh.
You're expecting the player to act as a DVR, rather than play a live event? This is roughly the behavior that VidyoReplay has (with a "go to live" button), and we get a significant number of complaints about that behavior as well.
I'm expecting it to act like DVR acts with live TV. I'm curious about the complaints about this implementation. Either way, it sounds like there are some significant usability foibles. Perhaps there are other similar systems which could be used as UX references?
Pausing and resuming live streams is beyond the limitations of current player technology.
If it's not possible for that button to do what the user expects, is there some way to remove or re-label it so that users aren't confused/disappointed?