"It's also unclear to me how exactly philosophy is better preparation. My own experience with academic philosophy is limited, but from what I can tell its main selling point is that it teaches you to examine things in a critical way that's very aware of the argument in question, the assumptions made, and the structure the argument sits in."This really depends on the kind of philosophy you study, which philosophers you're interested in, and how you engage with and react to what you learn.
Asking what you get from studying philosophy is a little like asking what you get from reading books. It really depends on which books you read, how you read them, who you are, what stage of your life you're in, etc. The same books might speak differently to different people, and even to the same person at another stage in their life.
The English-speaking world (and increasingly the rest of the world) is dominated by analytic philosophy, which is one lens, or perhaps a family of lenses, to view philosophy through. But there's also continental philosophy, ancient philosophy, Eastern philosophy/religion, Pragamatism, etc. Each of them brings something different to the table, and has the potential to transform the way you understand the world.
Sure, philosophy can help with critical thinking skills, but to limit it to that is to ignore the content of what one actually learns and engages with. At its best, it can't be reduced to just a bundle of skills, set of knowledge, or "selling point".