Walt's "confession" to Jesse came as a result of his need to "explain" things to him.
remember how Walt went on and on about how Jesse just needed to have something like Brock's poisoning "explained" so that he would see things Walt's way and understand why Walt did what he did?
this was Walt explaining to Jesse. that's why he wanted to meet with him in the first place in the square and Jesse backed out. it's not that Walt was wrong to do what he did, but that he hadn't yet had the chance to explain it to Jesse.
sure, it's convenient, and compressed for time, as this is a TV show and not a documentary, but one gets the sense, based on attentively watching the show and paying attention to textual clues, that Walt had been rehearsing what he was going to say for a while, and it all came spilling out in a moment of extreme stress.