Yes, this is how Fate Core is expected to be played.
I'm not going to quote a bunch of Fate rules, I'm going to answer the underlying question of how do you learn to GM this way and what's fulfilling about it. I think my story parallels the development of the gaming industry on this point - it's a journey in becoming comfortable with player narration.
It is a big shock to people when they come from trad games where the assumption is that GMs jealously guard complete control over all aspects of the setting, because the players having it would reduce challenge, surprises, etc. I started gaming in the early 1980s, so I too was brought up with that strict Gygaxian firewalling as my default gaming metaphor. As RPGs came from strictly adjudicated minis games, it's not a mystery as to why this was the case. For the first two decades of RPGs, this metaphor held iron rule.
The first change to it came for me with the game Feng Shui (courtesy Robin Laws), which gave the advice that hey, if the PCs are fighting in a pizza parlor, and one of them wants to pick up a pizza cutter and slash some goon in the face with it, they should just be able to do that. They shouldn't ask you, they should just do it, but of course if they pick up a missile launcher from behind the counter you call BS. I trained my players into this quickly by telling them "if you ask me, I'll say no. If you just do it, then it's all good."
This resulted in faster, more dynamic, more interesting scenes. And what was lost? Me rolling randomly to determine if there was a pizza cutter nearby? Not letting someone do something fun they'd just come up with because it's not the move I would make in their place?
So this crept into our trad gaming even in our D&D games (and by our, I mean mine, but also in general gamers across the world). Why waste time and take away momentum for no value? Now came games that started to experiment with that more. In the name of heroism and story, more and more games began to have a "hero point" like currency that would allow a player to affect the story by cashing one in and saying "that guy missed!" or similar. It was before Fate; I can't remember the first game I played that let you cash in a couple "hero points" to make a fictional declaration, but then that became a thing. "Here's my 2 poker chips, that guy who busted in the door is actually my long lost brother from my backstory!!!"
Now, this could derail the GM's sense of their own story. But in general, the 2.5 decades of accumulated GMing wisdom by this time generally weighed in on the side of "stop being so in love with yourself and your 'story.' Go write a book if you want to tell a story, run a game to have the players participate meaningfully." Early D&D modules "on rails" soured the player base against that; there was a AD&D 2e Forgotten Realms module where you basically follow gods around doing their thing during the time of troubles... So that, plus the fact that usually there was some currency wrapped around it so it wasn't just you spouting off about everything coming in the door, became very common and continued to erode the trad "no player narration" model.
Then you had games like Fate that embraced that to a greater degree. Actually, early Fate was more like what I describe above, requiring Fate Point spends to influence the world. Spirit of the Century is basically a trad game with an option to point spend for declarations and more "wiggly" ability definitions than many games. Subsequent revs of Fate went more that way. With Dresden Files the codified idea of players participating in setting creation was added.
As more indie games played with more and more player narration, games like Apocalypse World and the other Powered by the Apocalypse became very popular, coming at the game from a largely opposite direction - the GM sets up some loose scenario but then the declarations of the players are what mostly drives the plot. As that's become ascendant, Fate has continued - both in the way it's written but also just in the way people play it - down that path. Fate as written may not be as freewheeling as what you saw in that video, but that playstyle with it is certainly not uncommon. So you might be in one Fate game that's a lot more GM-narrated and one that's a lot less, based on the table involved.
I also enjoy exploratory play, which player narration definitely strongly affects. But you don't have to play every game/campaign/etc the same way. While I'm still up for a good trad sim game with exploratory elements, our group has also played Fate, Dungeon World, and Fiasco games where much more of the situation comes from the players.
It might be helpful to compare - my group likes Paizo adventure paths. We have run a bunch using Pathfinder, for example Curse of the Crimson Throne - you can see a bunch of detailed session summaries at the link. But we also ran Mummy's Mask using Dungeon World, and have summaries of that result.
From the player point of view it was fun, and what we lost in exploratory simulation we made up in the game moving somewhere around 50% faster than usual, and not being stuck in situations where "our characters didn't have a reason to be there" as we often do in trad adventures.
In a player narration heavy game, the GM still has plenty of role to play. They control consequences of actions, and also they subtly guide the situation via framing (the "who died in here" example). They get to focus more, in fact, on coming up with neat new things to inject than in worrying about the minutiae, though they do have to think fast on their feet.
In conclusion, Fate can be played more or less this way, and you can start out with just introducing some player setting-building and narration and increase it as you get comfortable; it's a big change to make and it's understandable you feel some culture shock. But take heart in that plenty of trad gamers have crossed that chasm before you, and whether it turns out to be "your favorite" or not, there's plenty of things to recommend it, even if just as an occasional palate-cleanser.
I'll be honest, I am not a FATE fan because I feel like it isn't written as narratively as it should be (full disclosure, I haven't read the newest stuff, but have played it up through Dresden Files). It seems to me like a trad game with a little narration on top, and I'd rather play a sim game if I'm going to wrestle with rules or a full narrative game if I want that. It feels to me like that group was indeed bringing a lot of playstyle to the table that Fate doesn't prohibit but also doesn't necessarily prescribe. But that might make it even better as a bridge game.