I would argue that to be a "game", the activity would require some sort of strategic thinking. I don't really put storytelling or imaginative play in the "game" category, or a footrace for that matter.
I agree with your view that "games" require some sort of strategic thinking, but I think that DM-arbitrated games
can involve this. Instead of playing the opposition via the rules, you just play the DM, by which I mean you use your social perception and persuasion skills to lead the DM to agree that what you want
your character to do is "reasonable", while what others (including the DM) want
their characters to do is "unrealistic". In other words, you try to divine how the DM thinks or wants the game/real world to work, and then you either parse your in-game intentions to fit that model or use persuasion and influence to modify the DM's model into something you can use more easily.
I have seen this done many times, and I recognise it pretty readily these days. I don't like it at all, personally, but I can see that some do, and they would logically be attracted to "DM arbitration" games as a result. Of course, not
all folks who like DM arbitration play this sort of "game" - some don't really play RPGs to "game" at all. I can see the attraction of that, but I find that the drivers to competition need to be way down or one or two folks can't resist a bit of "play the DM" - which I notice and get annoyed about. Hence, for me to enjoy pure sim/immersive stuff things like xp, levels, "character development" (in the specific sense of expected character improvement) and treasure hunting need to be marginalised or removed as elements of in-game power determination.