Performance has always been a big part of my experience at the table (I have a theatre background too!), to the point where I notice how often other folks prefer to keep it light or avoid it entirely. It's interesting to see which games "get out of the way" to facilitate performance. Like, D&D as a rules system doesn't provide any structure for roleplay, but because of that it can be a capable facilitator for freeform RP. One reason why I think shows like Critical Role and The Adventure Zone are so popular is because the players can just riff with each other without worrying about picking up the dice.
Contrast that with games like Monsterhearts, which are all about social interaction and has a lot of rules to facilitate it. But introducing those dice in the midst of roleplay "performances" is a disruption that D&D can more easily avoid. Or for a really egregious example, look at Blades in the Dark: every action roll in the game is the result of a conversation that goes back and forth between player and GM several times. "Performance roleplay" is possible in both these systems but both present unique challenges too.