I believe you have brought up another final important point: The cast was responding less to Pence’s presence, and the more to the audience’s response to Pence’s presence.
It was disrespectful for the audience to boo Pence upon his entrances, and disrupt the performance repeatedly whenever it was remembered that he was in the room. It was not disrespectful for the cast to respond to what was happening, and that’s what Dixon’s speech was. It was a response to what was happening. It wasn’t an attack on Mike Pence. The audience attacked him, and it was the audience who was “ranting” the whole time. The cast sought to rectify that.
Personally, I think it would have been deeply irresponsible for the cast of such an important production to ignore the volatile situation going on within their theatre that night. Dixon’s speech fought to put the reaction to Mike Pence’s presence into perspective, and he asked people to stop booing so everyone could be heard. The speech had a goal to unify, and to ask Mike Pence to not hear the booing as hate, but as fear that certain marginalized people do not feel protected.
Dixon respectfully expressed why people were upset without validating it in hatred and allowing them to continue the disrespect, which is why it was important that it was done in front of the audience. He put a stop to the loutish behavior that they couldn’t previously stop from the stage and voiced the concerns in words of compassion that related to the story they had just presented within the show. As far as post-curtain speeches go, it all makes lots of sense and ties together very nicely.
The cast had a responsibility, as it were, to wrangle the atmosphere of their theatre that night. To do otherwise would be for them to say, “Well, we performed our show. Now lets go take our paychecks and bury our heads back in the sand and let the booing continue tomorrow night.”