Right. So who opens the safe will affect the contents of the safe.
Your phrasing implies that you think there are already some contents in the safe, and that we are changing them. This is not the case. In such a play style, we are not exploring a pre-existing reality—
we are writing it as we go.
Put another way, we aren't reading a novel, not even a "choose your own adventure" book where all the options are already fixed in ink on paper. There is no safe, and the safe has no contents, until we decide what they are. We're writing the module or adventure as we go through it. As for which skill is used, there has to be some plausible fiction for how the characters get the safe open. And as I've pointed out a couple times, how they go about it will have different potential consequences/fallout for the fiction.
But also, it isn't necessarily who opens the safe that determines the contents. No, that just determines the consequences/fallout of the opening. It's entirely possible that one player puts forward the stakes on what's in the safe, everybody agress to that, and then they nominate somebody else to open it! The dissociation is even worse, good heavens!
But seriously, that's absolutely true. There is no
necessary association between the character doing the check and the outcome. It's
all negotiated by the participants in the collaborative process of generating the fiction. The negotiation may be collaborative or adversarial (as
@Manbearcat's quotation of the relevant text from Blades in the Dark shows), but the important point is that we're generating things as we go.
But the characters presumably don't think this is the case, yet the players know it to be so. This sort of system causes almost complete disconnect between the decision making process of the players and the decision making process of the characters.
Yes, there is a disconnect. It's a different play style, a different way of generating a fiction, to the usual exploration of a prewritten situation/scenario. That's all.
Hell, why would the characters even debate who opens the safe? They certainly cannot know it would in any way or form affect the contents! In universe it would make perfect sense for one character to say "Go ahead, you do it, you're good with locks. But I sure hope it's those papers in there rather than just some pointless gold you're always after!"
They debate it because
how they open the safe
matters, possibly more than what's inside. If the party needs to get Jane's safecracking skills up, have her do it (maybe the master safecracker is going to retire soon, or maybe that character's player is leaving the campaign—a very dissociated concern). If we don't care about raising an alarm or leaving evidence, have the smasher do it. If we're really worried about getting that safe open without complications, then sure, have the master safecracker do it.
This obviously doesn't bug you, and good for you. But it would bug me massively.
That's totally cool. You have different preferences in play style. You like to follow things forward from established premises. The approach I'm describing involves quite a bit of logical backtracking and filling in. Heck, Blades in the Dark has a full-on "flashback" mechanic where you can literally go back in time to fill in detail...so long as it doesn't directly contradict what's been established, anyhow. Me, I love playing that way. But I love following forward too, it all depends on the game & the group.
It is not just colour, it affects the odds massively. In fact GMs determination of the difficulty affects things way more steeply in BW than in 5e D&D, and it is far easier to make things virtually or literally impossible. Why on Earth would such a huge impact be given to simulationist measures such as safe quality, which has basically nothing to do with what you're actually intertest about, that being the beliefs and desires of the characters?
Oh, sure it can affect the odds (very little in Apocalypse World, quite a bit in Blades). Balancing odds with the stakes is a big part of the fun of playing Blades! Well, for me anyhow.

And Blades gives you lots more tools than quality of items to juggle and jigger in the process of working out those crux moments. I love it to bits.
Edit: Fixed a typo.