> 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.

That's definitely a big (and important) part of it. There's an unfortunate dearth of critical thinking and examination of ideas these days. I have theories as to why that is, but it's out of scope for this discussion so I will abstain.

Another big part of philosophy as well is to familiarize the student with the big ideas with the hope that they can be further refined, to drive both the individual thinker forward, and hopefully humanity as a whole as well.

For example, I think a hugely valuable part of philosophy training is the study of ethics. So many people (software engineers and companies included) don't really stop to think about the ethics surrounding what they are doing. This is a mistake, IMHO. I highly recommend reading about and understanding Immanuel Kant's work in this area, especially his categorical imperative and hypothetical imperative. Comparing/contrasting that with Jon Stuart Mill's Utilitarianism, and conducting various thought experiments makes for a good time (you can tell I'm quite the partier :-D ).