Making screenshots and screen recordings of Android phones

If you've found yourself needing a quick screenshot or recording of an Android device, you've discovered that it's damn near a secret how to do it: download this terminal app, run this cryptic terminal command, and this one, and this one.

Why isn't there an app that just lets me plug in my phone and click one, as in 1, single button?

I first made this for my team mates at Android, but figured more people could probably use it. And it turns out that they could.