Why isn't this player more engaged in the game? Why is he only engaged when he is rolling dice for his own character? Can he not find enjoyment by observing the actions of his fellow players, sharing his observations and suggestions, maybe even offering to handle part of that menagerie so the "menagerie character" doesn't have such a workload?
Is his only option to go outside until it's his turn?
Isn't that kind of self-centered? "Oh, golly, it's not my turn, so now I'm bored and it's time for me to take my marbles and go home."
Yes, I see your point about mintes being a zero-sum game. But that's only true if each player sees them as "my minutes are cool, your minutes are wasting my time".
As for me, when other players are taking their turn, I'm offering suggestions "hey, why don't you pull a Legolas and go jump on the troll's back?" Maybe it's to be comical, maybe it's to make a real suggestion in case that player hadn't thought of it. Maybe my comments contribute to the specific encounter, or to the game, and maybe it's just goofing around. But regardless, my involvement is contributing to my fun all the time. And the other players in my group do the same thing. We're all having fun, even if it's currently some other player's minutes that are taking away from my own minutes.
In this case, it's not a zero-sum game. It's a 60-sum game: Every hour has 60 minutes of fun, and every player is enjoying all 60 them.
I hate to say it, but if any player's personal world view is that "when my friend is having fun, but I am not rolling dice at this exact moment, then he is stealing my fun", then that player might want to adjust that world view.