I feel like I'm reasonably good at picking at a game on the gameplay level, as per what works and does not and why and surface videogame essayist stuff like ludonarrative dissonance (or the rare examples of ludonarrative harmony).
I may offer you my finest insight into video games such as "Lara Croft has some sort of father complex going on" and "Shadow of Chernobyl is unintentionally about life in the collapse of the soviet union" which even by my own admission feels shallow and trite. You watch someone like Jacob Geller or Noah Caldwell-Gervais and they have fascinating things to say even on games you wouldn't expect it, like NCG on Quake.
How do I become that knowledgeable? Interesting? Analytical? about video games?
Not to cheapen any other post here (because I think they're genuinely good) but don't forget there's nothing wrong with those two examples you gave. You don't have to get deeper with it.
You could make an interesting videogame essay about the aesthetics of collapse on shadows of Chernobyl and how that's based on the Soviet collapse. That's insightful! You're trying to take this expression of a concept and dig into why they used it, their reference points, why they thought it was important to make etc. That's already interesting!
That /is/ being analytical and knowledgeable. Not just with games but with any art. If that's something about the game you think is interesting or resonated with you: dig into it. Might end up being a shallow hole, sure, but not everything worth the time is buried deeply.