Yes, it's a vent-piece, but I don't think it's cringy. For me, there's a huge difference between an artsy vent-piece about some vague "darkness in my heart" or "monsters in my head," and a vent-piece that uses a fun allegory to explore something acutely and presently relatable to a large chunk of your audience. The "darkness in my heart" thing is 'cringe,' I think, because it sort of cheats its way to being relatable; it makes itself moldable to almost everyone's sense of internal, personal, aesthetic. Some people can relate to it in a more acute and defined manner, of course, but a lot of the time it's just the fact that everyone sort of wants to have some hidden darkness in their heart.
This piece resonated with me on a much deeper level because your reader can tell you're not trying to be that deep. I have been in this situation. Literally. Right down to being not honestly that great at Solitaire, but still using it to procrastinate. So hush your worries about cringe. It really really works.
I have two main 'complaints' (not really the word I'm looking for, but you know that struggle well enough I'm sure) about how this piece is written. These are your descriptions of the game itself, and the way you defined the allegory.
Second first, because I think it's a much easier problem to solve. My issue is with the word "moral."
…while a palpable, clearly visible question of morality is put before you…
This immediately confused me, but upon reading further I think I know what happened. You threw the word "morality" in there while drafting as a placeholder, but then it led you down a path and clouded your rear-view, and you had written the trolley problem into your piece about procrastinating on homework before you realized that those don't really connect that well. I think the word you were aiming for is "strategy." Your entire preamble was about the strategy of the game. In your allegory, winning the game is getting your homework done. Whether or not you win the game and whether or not you get your homework done only affect you; they require selfish strategy. In contrast, the trolley problem is about the morality of how your actions (or inactions) affect other people. Change that word, and it will force you to change your trolley problem thing to something much more self-centered.
The other problem is as old as literature, really. Board games and card games, if we're honest with ourselves, are most clearly, efficiently, and succinctly represented using some mathematical notation or another. Keeping track of a detailed game-state that lives in the visual just doesn't really go with the neurological flow of this thing we call reading. And authors have been bumping up against that problem every time any of us have wanted to involve a game of chess, or connect-4, or hell even a videogame in our stories. Math is the bridge between writing and a visual game, but not a very good one; equations don't make for engaged reading.
There's… not really much you can do about this, to be honest, but one thing you can do is make it so that we're only trying to keep track of one game-state, and not two. There's a point where the narrator begins using second person to describe a hypothetical game of Solitaire. But in the back of the reader's mind, the Procrastinator has just started a separate game as well. So we're kind of floating above him having a tutorial of sorts while the 'real' game is paused below us.
I would much rather this be integrated; it even provides you with more chances for the narrator to be frustrated with the Procrastinator if you're describing what he does vs what he should have done. How a short-sighted decision at the beginning of the game caused him to lose down the line. This would also help the story have a more secure flow of time.
In terms of vibes, I really like it overall. I like the rambliness, I like the pretentiousness, I like the references to real-world Solitaire theory and philosophy.
I think you have room to expand a lot more on the 'voice in a head' bit. I know because I know exactly what you're trying to describe and how it feels. I feel like I'm the powerless talkative conscience, and my real brain and my real body are as likely to ignore me as they are to listen to me. But I don't want you to try to expand on this by force as you edit; I think if you have it in the back of your mind while you work on the other stuff, it'll come through nicely.
Unsure how to conclude, as I realize I've typed almost as many words as the original piece. Just don't be too hard on yourself for the fact that, by the nature of venting, not all of the words you used conveyed what you were trying to convey. That's called being Human.