Actually, I think people have made systematic attempts to teach it. Those attempts were named 'Zen'

Ah, right. I should've said, in the self-help field, or more precisely, in the subset of the self-help field that doesn't appear to descend into irrational madness. Silly of me to forget Zen, since I've actually studied it -- and not just in the "read books and practiced at home" sense. I'm just reluctant to strongly recommend other people study it, because it sounds too mystical or "irrational". Perhaps I should change that. (My reluctance, I mean.)

Then I read something you wrote, about observing your own reactions, and I was enlightened: the purpose is to put your mind in a baseline state so that you can observe all the things which pull you away from it, and learn how to deal with them. (First acknowledge, then suppress them.)

Almost right. You don't suppress them, you let them go. Suppressing them would strengthen them, for the same reason that "not thinking of a pink elephant" doesn't work. And it's not so much a baseline state, as having a task upon which to concentrate. It doesn't matter what the task is; it's just easier to learn if the task doesn't involve any activity for you to get caught up in thinking about. Once you learn to get into the state, it's possible to keep it while doing other things. For example, the Zen center I attended in Dallas did walking meditation in between sessions of sitting. It would've been very hard to start with walking meditation, but it was relatively easy to stay in state during it.

Suppose you recognize that you have a song stuck in your head, and consciously decide that you don't want it there. Does that decision have any effect?

In my experience, none whatsoever. They last for days, and I've never found anything that gets rid of them, except replacing them with something else... which usually requires an external input, rather than any mental activity.

If you later find yourself thinking about abortion or gun control, and your conscious mind declares "politics is the mind killer, I will stop thinking about this", does it work? I believe that these are the same skill, and that meditation, if done properly, builds that skill.

Nope. Doesn't work that way. You can't decide not to have thoughts. All you get to choose is to refocus your thoughts on what you intended to focus on. Refocusing and detachment are the skills you get from meditation. (Detachment is also useful for mind-hacking, because it lets you separate observation of your response from engaging in the response.)

Think of it this way. Your mind is a table-driven state machine, constantly responding to the environment and to its own fed-back outputs. Normally, when thoughts come up, they loop back into the state machine as input, driving feed-forward behavior. You think, "this sucks" or "I'm bored", and that then feeds back into the machine and makes you think MORE about how much it sucks or what you could be doing instead of this boring task.

The skill of detachment is being able to notice that thought as a thought, and NOT feed it back into the machine. You refrain from "following the thought", and simply continue on your task. You're training a general response to all thoughts as "ah, that's an interesting thought, and now I'll continue with what I've already chosen to do."

What you have to understand is that fighting or trying to suppress the thought is just as bad as becoming immersed in it, because you're still creating a feedback loop, despite it being in opposition to the thought. You're still enmeshed in action-reaction, instead of remaining focused.

The skill you develop is also similar to something pickup artists call "cutting the thread" -- when an unpleasant topic of conversation arises, or somebody says something that leads away from where they want to go, they simply acknowledge the statement in a way that makes the person feel heard, and then continue leading the conversation where they want it to go. They don't feel obligated to either follow the thread, OR argue with it. (They also use the term "non-reactive", which is a good general term for this idea, I think.)

Non-reactivity is useful in that it strengthens willpower. In my work, though, I don't emphasize it as a way of developing willpower, but as a way of applying techniques that reduce the need for using willpower in the first place. That way, it has more leverage. You only need to be non-reactive enough to apply a technique, rather than striving for 24/7 nonreactivity.