Learn English – nice to see you vs good to see you

[verb] one's way {{in | into | to } [target] | through [hindrance/obstacle]}

is a general pattern where you can substitute different verbs, like "force", "claw", "push", "work", "beg", or really anything that is clearly understandable in the context.

It means, roughly, "Attaining [target] by [verb]-ing, thereby creating a path or entry into [target] (or through [obstacle]) for oneself, where a path or entry wasn't available."

The worm cheerfully munched its way right to the center of the apple. [There was no path to the center of the apple; the worm created one for itself by eating.]

She slept her way into the movie industry. [She had sex with directors or producers in exchange for getting roles in films, thereby creating a way into the industry that was not otherwise available.]

The usage "don't you have to push your way in" is clear: "don't you have to get into the game by means of creating an entry for yourself via pushing?"

"in" is correct, because that's the preposition that is used for indicating participation in a game: being "in the game".

"Push" is also appropriate, because it has figurative meanings, one of which is forcing or asserting oneself. (E.g. an overly assertive person is called "pushy" and "pushes around" other people).

I think you're overanalysing this one. Obviously native speakers often make "slips" like this, especially in casual conversation or Q&A website comment posts.

In the case of X mistaken for Y, I imagine almost every competent native speaker would say for is "correct", but many if not most wouldn't see anything seriously "wrong" with as. A (probably, much smaller) number might not see much wrong with with.

This is an example of a comparatively minor slip. On a site like ELL, for example, many users are native speakers, and many aren't. As it happens, we have contextual information for the specific comment that sparked this question, making it highly likely that the writer is in fact a native speaker. But even if there were no such evidence, the fact of the usage itself would be no real reason to suppose otherwise.

This shouldn't be taken as a general principle for all "misused" prepositions, etc. Some errors are far less likely from native speakers, and far more likely to be seen as "giveaways" indicating "non-native".