This is a great example of what we can call "selection games."
The defendant is being selected for either a mental health institution, a death sentence, or life in prison, and he has preferences between those options. He's defended on the basis of his behavior—what that behavior indicates about his mental state. So there's an obvious attraction to behaving in ways that lead to the selection game outcome he's after.
And, briefly, it's clear that we can never know for certain whether someone is sane or insane—what's inside their head is a black box, and we don't even have very clear ideas about what sanity is, to begin with. All we have are these outside signs, cues, etc. And we get impressions about what "crazy" looks like from life experiences, from watching movies and television. And then we act those out.
Big gaps in private vs. public behavior is one of the main ways you can actually identify what's "playing the game" and what's "real"
And I love the way the video emphasizes, with the Parkland shooter, that the more he says up front, the narrower his options are down the line. That you wanna leave options open, keep the most paths in the garden of forking paths available. This is sometimes called "intrinsic empowerment" in AI. All else equal, maximize degrees of freedom. It's why exercising your 5th Amendment rights is important—you're keeping options for possible play strategies open, til you get a lawyer. But when you claim one kind of thing ("demons, voices") early on, you gotta now stick with that strategy. It limits you.