That only works if the page is relatively static. Add in any functionality that requires a round-trip to a server, and things get immensely more complex.

No, they don't. There's a benefit in being able to snapshot the current state of a dynamic application.

But what happens when you want to meaningfully interact with it? It seems kind of useless to me if things are broken the instant I try to do anything.

You can press the Refresh button and suddenly things fix themselves. The utility is in having a snapshot of the site looking exactly as when you left it, with all the benefits of it not being rendered to a image, and with a high probability that at least some interactions still work.