I do think Dogs conflicts allow you to be a little more fluid with some of this than BW’s DoW, and part of that is, as Dan says, the turn structure (and/or the way just pushing dice forward lets you signal “this is a raise” without having to interrupt.) That aspect does break down a bit when you have to bring in traits or relationships, though: that requires more mechanical talk.
I’ve seen some of those “tradition” elements come out very strong in games where it was something the players brought to the table. I suspect that it’s very difficult to inculcate a whole worldview at a “tool” level: it probably does take a whole game. But if you’re comfortable with the idea that your traditions or values or what-have-you are going to have to be things the players already have some reference for or interest in, it’s possible to build a game which encourages them to come out in play without having to be exclusively about that.
(Did that make any sense? I’m decidedly not saying “system doesn’t matter,” obvs, maybe more “system can’t do the whole job but I bet we can design systems to give a boost.”)
I really really care about worldviews besides my own, especially in terms of history. So it’s something I’m going to push whether the rules point me that way or not. But besides Pendragon (which is, approximately, actually pushing a fictional and not even realistically-fictional worldview) and Clay, I’d say Dogs is definitely in this ballpark, it’s just doing more of Kit’s “don’t tell me, but you have to know” thing. There aren’t rules for believing in or silently disputing the tenets of the Faith, but you’re clearly pushed towards those options.
BW Grief and Greed are maybe related too, on the opposite end: they’re mechanically heavy, and you don’t have to assimilate them in anything approaching a “method” format, but if you don’t do the thing, something happens. Might be worth another look at Blossoms for a more historical-worldview version of that, but it’s been too long for me to say anything strong on that score.