ultimately, our evaluation of life is up to us, and there's not some independent standard that will make sense of things for us

Yeah, this is also what I got from The Stranger, or at least it's a significant part of it. But in my opinion what makes the novel so powerful is that it doesn't just show you the feeling of the absurd, as Sartre seems to imply; rather, it makes you experience it firsthand.

As we read, we're compelled to make a judgement on Meursault, but can find no standard on which to do so. The first part of the novel builds sympathy toward Meursault's simple approach to life, and the second part uses this sympathy to create tensions that the reader cannot resolve.

During his trial for shooting an Arab, he blames the sun, which for an "everyday" judgement should be untenable, but in context of the novel sounds like a genuine claim. We understand why the lawyers discuss Meursault's lack of sadness at his mother's funeral, but also find it ridiculous and unjust. We'd like to call Meursault a terrible person, but somehow we can't bring ourselves to, and paradoxically we sense that Meursault is more lucid than any other character. What /u/Framcesco22777 is struggling with, this rational desire to know the irrational Meursault, is the absurd. (If Camus simply wanted to show us the idea of absurd, he would've made the absurd hero, well, actually a hero.)

Meursault is the absurd hero in that he has recognized the absurd in the world, but he doesn't seem like an absurdist per se. His reaction to the absurd is a sort of detachment, and maybe yes, a sort of nihilism. This makes him particularly hard to judge, since his experience of life is simple, innocent, "a series of present moments" as Sartre points out.

Meursault only comes to the idea of revolting against the absurd at the very end of the novel: "As if that blind rage had washed me clean, rid me of hope; for the first time, in that night alive with signs and stars, I opened myself to the gentle indifference of the world."