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?
You need to write down your thoughts. If you don't like writing then you need to talk to others and discuss the themes together. You'll never notice everything because you have your own lense through which you see the world, and that's why talking to others is useful to reveal the lense that other use to understand the media they consume. You will hopefully each be able to scratch away part of the underlying themes and such that go beyond the surface level stuff.
I'm incredibly bad at it as well. I'm reading books along with shelved by genre (Austin Walker and ranged touch people) and I'm never able to match their analysis. It's deeply frustrating sometimes because I really struggle sometimes to make the connections like they do, but they're like professional media critics, and I'm just some dude who never really dipped my toes into this until the last decade or so. Just gotta keep trying and maybe accept we will never be able to decode all the bits in our media.