Well, in a style like this, there are lots of things the GM knows in advance. He doesn't know what is going to happen though. He just knows that the brother is dead. Maybe when the player gets their he tries to resurrect him. Or maybe the player character goes on a murderous rampage after. There are all kinds of places that could lead, that the GM does not know. But you are right he knows that a successful search for the brother would yield knowledge of his death.
By your description of the PbtA approach, it sounds like the setting detail (the brother being alive or dead) is being baked into the player setting that as a goal for the relevant check. If that is how things are done in PbtA, that is fine. People are happy who play those games. My point is, in a sandbox, framing this way, is setting up the outcome, and something you wouldn't do. In most sandbox games a player saying he or she wants to look for their brother isn't going to be distilled into one roll or action. It would like be a number of efforts at tracking down rumors, clues, etc. And there would simply be no assurance that he is alive at the end of that (nor would most people in a sandbox consider an outcome where he is dead as futile (if anything they might be suspicious that it sounds overly dramatic, especially if it involves any of the details you mention above, but I like drama in my sandbox). Also this is just one possible outcome. What makes it exciting is it is an unknown on the player side. One possibility is he is dead. Another possibility is he is alive and waiting there to meet his brother again. Another is he is alive but filled with resentment towards his brother. Or we could even take a page from Death Duel and have him find a coffin upon ending his search, only to later discover his brother faked his own death and has been living wretched existence as nameless wanderer later on. There are all kinds of potential outcomes to "I go look for my long lost brother" in a sandbox. But as a player in that kind of campaign, I don't expect to shape the outcome. I get in other styles of play, and in some RPGs, the expectation is different, and that is fair (and maybe there is an OSR adjacent sandbox style that does that as a lot of the PbtA fans seem to be interested in Old School stuff recently). All those styles are fine by me. But what I am describing is the more OSR rooted, sandbox and living world approach. In this, I really do think the brothers status as alive or dead, would be something that players would expect the GM to decide, and they wouldn't see that decision as infringing on their agency.
In PbtA (and other similar systems) players get to set goals, yes. The goals must be coherent with existing established fiction, genre, etc., although there isn't really a specific process for other game participants to reject what they consider inappropriate/contradictory (presumably this would rely on table etiquette and such, but I have never seen it be a big issue in play). In PbtA SPECIFICALLY, the actual player inputs are fairly indirect, players don't really get to simply say "such and such is so." Going by Dungeon World, which I know far better than the others, there are something like the following avenues for player setting input:
1. GM questions - the GM is directed to 'ask questions' as a general technique of play, but especially at the start of a new campaign. This is fairly loose, but it implies that the players will inject certain fiction into the game at this point; IE if a player asks where his PC might come from, the GM would turn the question back to the player to answer, so the player might thus establish the existence of a 'Steading' within the setting.
2. Bonds - players write bonds for their PCs, both at the start of the game and on an ongoing basis. These establish relationships, which could in turn establish fiction to an extent, though these bonds are rarely over one sentence in length, and usually declarative.
3. Spout Lore - this move causes the GM to tell the player something his PC knows on a given topic. The GM may be constrained to make this information 'useful' and 'relevant to the situation' on a 10+. Creative use of this move can constrain the GM pretty heavily, but GMs can also respond to the PC's move with a move of their own! So it tends to be a pretty limited avenue for player injection of lore.
4. Discern Realities - Like Spout Lore this can be used cleverly by players to, at least, induce the GM to elaborate on a given location. Again the GM can respond with a move.
5. General Fiction - PbtA games in general are 'fiction first'. While players don't have license to simply invent any old thing, they can develop fiction by means of action declarations. For example a player might state that he is attempting to "Push the Orc off the slippery rock." and thus establish the rock's slipperiness, or perhaps even establish what the floor is made of. The GM might counter this kind of thing with something like a Reveal an Unpleasant Truth, move "Oh, that rock wasn't as slippery as you thought!" although this would normally be in response to a 6- on whatever the check was to do the pushing it could simply be a response to something that seems out of line in terms of the existing fiction.
Notice that none of these absolutely gives a Player carte blanche to establish fiction, at least not without the active cooperation of the GM. The GM's agenda and principles of play do, however, indicate that the GM needs to give the players a good bit of leeway.
I would note that, in other respects, DW has some 'sandbox like' features, in that the GM establishes 'fronts' (major elements of meta-plot). These would include 'dangers' and also 'omens' which certainly can work in a very 'sandbox like' way. That is they can simply be things which lie in wait for players to interact with them, and hints as to these things existence, which is pretty stock sandbox fare. While the GM is supposed to have 'holes' in her maps in order to accommodate players development of fiction, said maps could well be pretty extensive too. So, the approach to the established fiction will dictate the style of play to an extent.
DW, at least, is less about the players getting to establish facts about the world as it is about directing play into paths which engage with fiction that is related to what the players would like their characters to experience. So, for example, the "your brother is dead" thing probably wouldn't happen in DW because the GM is a 'fan of the players' and his job is to keep giving them material where they can both shine, and get into deeper trouble than they are already in. Thus killing off an avenue of development in those directions would usually be contrary to the GM's defined agenda. Of course it might still come out in play as an outcome, but in that case the action would move on to a more interesting/challenging/dangerous stage (IE maybe you now have to travel to the land of the dead to get your brother back!).
