"It seems like you're almost completely left to your own devices regarding curriculum, testing methods, even classroom texts used. Is that common? I'm curious about the variations across programs/schools regarding institutional oversight."
Let me tell you about my experience. I worked at 2 hagwons before I found a uni job. At both hagwons, they gave me the books to teach and had a schedule of what pages to teach each day. It was really quite simple and just required a few minutes of prep time. I would say that most hagwons are like this, BUT, one time I did have a friend who was literally given no book to teach from. She had to make her own materials each day. She had a quite stressful year. This is perhaps a question to ask your potential boss. You will have a much easier year if books are provided for the kids.
At the university, it can totally vary. In many cases, you are given a book and expected to teach it but I've never been given a schedule of what days to teach what. I've also taught classes like writing, or advanced conversation where all I got was the class title and had to figure out the rest. In one case, I did a 2 week long summer camp for kids and was literally given no materials to teach from.
And at public schools, it varies (from what I've heard, I've never worked at one) as well. Sometimes you have a book, and a co-teacher, and a plan. Other times, no one cares what you do, so you can do whatever you want basically. This is actually not so bad at a public school, as compared to a hagwon because you usually only see the kids once/week (if not less!), so you can recycle your one lesson for every single class that week. At a hagwon, you'll see kids 2-3-4 times/week. And in most instances at public schools, you will have to plan a winter and summer camp, by yourself, full of fun things to do.
As far as testing goes:
Hagwon: usually a 1-1 speaking test, where you have to give everyone above 80% to keep the parents happy.
Public school: usually no testing. The Koreans teachers would never trust you with this. If they do, you might make up 5 or 10% of the final exam questions based on the material you taught.
Uni: it depends. I have complete freedom to administer tests and final grades. At other unis though, they have a common exam. My (totally untested) theory about this is that higher levels unis have more standards, and therefore common exams. The lower level unis don't really care and so you're left to your own devices.