Since there is a pretty powerful spring compressed inside the clutch, it is actually a fairly delicate process to remove. The first step is to remove the six bolts you see in the photo. I soaked them in penetrating oil, and they were still very tight, and when trying to loosen them, the whole assembly turns. I finally broke down and bought a cordless impact driver (and as I learned, not to be confused with an impact wrench) that works with batteries I already have. I got the impact driver because I figured it would be more useful in the long run than a dedicated impact wrench. I looked at the torque specs but they didn't seem that different. Anyway, the impact driver didn't budge the bolts. I was somewhat dejected, but I decided to try removing them manually one more time and they came right out. The impacting may have loosened things up.
This is the tricky part. You actually have to remove every other bolt and then insert different longer bolts into those holes, along with some spacers (I used sockets). You then remove the three remaining original bolts. After that, you slowly remove the longer bolts in unison. This allows the spring to decompress gradually. It looked like this:
I eventually got the thing out, and things were looking good.
Until I cracked it open and discovered the friction surfaces are covered in rust.
I'm not exactly sure how I should proceed. I'm inclined to sand/buff the rust off and give it a try. The actual clutch disk is well within the spec for thickness. One other question is how to tell if it has been contaminated with oil. I think it might be one of those things that is obvious when you see it, so I think I'm good there.
Now I need to decide if I want to remove the flywheel. I probably should in order to clean everything out, but it's a pretty sensitive process and I really don't have to. We shall see.