I’ll comment on your first sentence only, because I am not sure I understand the second one correctly. Could you clarify what do you mean? Indeed, the distinction might be blurry. I might have overly narrow view of politics, but it seems to me that while boundaries between them are not clear, the “central examples” seem to be different. I’ll try to present an example of what seems to me an archetypical example of politics. I do not claim that it describes all political discussions everywhere.
Imagine a parliament or some other group of people each of whom is of some political ideology. Suppose they have an issue in an agenda. If the issue is uncontroversial, everybody just uses their common sense and moves on to the next issue. However, if an issue at hand is controversial and complicated (where there is no clear and easy solution), then self-identified blues often default to doing what blues are supposed to do and self-identified greens default to doing what greens are supposed to do. Therefore some people would probably say that politics are best described as a clash(es) of ideas or a clash(es) of ideologies (e.g. there are many books with similar titles, also compare this to the clashes between candidates before the elections). Debates between people supporting different sides of these clashes are possibly the most central part of democratic politics. How exactly these debates are won or lost is of secondary importance (a good politician is one who can win such debates).
Game theory can be useful here e.g. to determine what concessions should a group of people make to strike a deal (possibly metaphorically) with another group of people to obtain a stronger position in a debate (by e.g. obtaining a winning majority). However, it seems to me that in this case game theory is used solely to reason about the level 2, and not about the level 1, i.e. it is used to determine how political actors would behave in such and such situation. However game theory seems to be rarely used for reasoning about the object level (level 1) (of course, the situation is probably entirely different in the field of foreign affairs). Take any polarizing issue that everyone has opinion about (what issues are considered polarizing might depend on the country). For the sake of concreteness take some issues that are controversial in the United States—abortions, gun policy, etc. These topics are definitely considered political, but it seems to me that they are rarely (if ever) thought about in terms of game theory. Of course, game theory might not be very applicable here. Or it might be. There might be even some game theorist somewhere analyzing these issues themselves, which seems to me to be distinct from analyzing related political clashes concerning these issues.
Of course, in reality it might turn out that level 1 and level 2 are too closely related to be analyzed separately. In that case the aforementioned distinction between using game theory at different levels might be unproductive. Also, I am definitely not an insider of politics—maybe politicians and experts who advise them use a lot of game theory for object level reasoning (i.e not about other political actors). However it doesn’t seem that way from outside.
I must also admit that I made a distinction between politics and game theory before I had a clearer idea why they seemed different to me. Therefore in this comment I mostly tried to find reasons why it might be so. I didn’t search for similarities and reasons why these two areas should be thought of as closely related. Due to this asymmetry my post might not describe a complete picture.
edit: I didn’t downvote you. I don’t know why somebody else did.
It seems to me that the issue could be made clearer by making a distinction between three different levels/types of politics.
The territory itself. Not the whole territory is relevant here. Let’s call relevant part preimage of politics . More about this later.
Methods of thought and style of thinking commonly used in politics. It seems to me that there are some issues that are understood as “archetypically” political (e.g.elections, election campaigns) and other things are deemed political (or not) by whether they are thought about in a manner similar to aforementioned archetypical examples. Thus it seems to me that the extension of what is thought about as politics happens at this level. Territory that is “covered” by this could be called preimage of politics(by analogy with mathematics), or “topics that are usually thought of as relating to politics”. One should note that in different contexts preimage of politics could be different. E.g. for a member of parliament, a journalist, an economist and an average layperson the sets of topics that they approach using “methods that are used in politics” might be different.
3.Political science, which studies the level 2.
It seems to me that when people caution against politicizing the issue, they usually mean not that the issue itself (level 1) should not be thought about, but rather that the issue shouldn’t be in the preimage of politics, i.e. methods of politics, style of thinking that is usually used in politics are probably not the right way to approach it (level 2) and other styles of thinking (e.g. economics, game theory) should be used instead. An adage “politics is a mindkiller” similarly cautions against the fact that a habit to extend the area covered by methods of politics seems to be self-catalyzing, i.e. once you politicize one topic/issue A you might be psychologically more inclined to politicize issues B and C. And after you “find the correct way of reasoning” about the topic using the methods of politics, you might be less inclined to try other approaches which are likely to be more productive. Once you politicize most topics it might be difficult to think about stuff in any other way, i.e. your mind might become metaphorically killed.
It seems to me that “politics is hard mode” is mostly about level 1, it captures the intuition that many issues that tend to be politicized (happens to be in a preimage of politics) are non-trivial. However “politics is the mind-killer” seems to be more about dangers at level 2, what kinds of approaches are used to a specific problem.
(English is not my first language, therefore I apologize in advance for all mistakes. Feel free to correct them)