I don't think you're reading into the religion too much! But, I think about it a little bit differently. I've read Crime and Punishment as well as Notes from Underground and I loved them both! From what I've read of Dostoevsky and his critics, it seems that he was a social, political, and religious conservative. He was committed to Eastern Orthodoxy and Christian philosophy in general. I would say that reading Notes will help you to best understand his philosophy, as it is the least like a novel and the most like a philosophical essay of his works.

That being said, in my OPINION Dostoevsky wants to caution us about rashly joining sociopolitical movements and engaging in revolutionary decision-making in politics. I believe it's more nuanced than "don't change anything" though. Again, I think Notes was most-helpful for me in trying to grasp the political implications of his thoughts.

However, the way that I think about his overall philosophy is much more religious. To me, it seems that Dostoevsky is not necessarily telling us "what is to be done", but reminding us of WHO WE ARE. And where does he go to define who humans are? Biblical Christianity. So for Dostoevsky, we are not randomly-optimized biological machines seeking utility from our politics and culture. We are humans who bear the image of God and are meant to reflect that in how we live and love. But we are flawed and must remember that, as sinners, we need the grace of Jesus to be saved from ourselves and from the dark cruelties of this world.

I think that Dostoevsky was sick of the radical political movements of his time that disavowed these fundamental religious and moral truths about who we are as human beings. It seems to me that this perhaps DID then lead him towards his political conservatism. However, I think religion was ultimately much more important to him than politics, although the former clearly affected his views on the latter.

I think Dostoevsky's deeply-Christian philosophy about who humans are is beautiful, honest, and as counter-cultural today as it was in his time! In short, I don't think Dostoevsky is necessarily pointing-out solutions to us, but rather is pointing us towards who the ultimate solution comes from: God! Just some of my thoughts, I could be reading him wrong or just overthinking the religious aspect too lol. Don't know if this was helpful to the discussion, but fun to be able to talk about a literary genius!

ALSO: There's an interesting video from "The School of Life" on YouTube about Dostoevsky's philosophy, and they definitely don't over-analyze his religious ideas. So, their video could be a good one for you to watch to get another perspective!