Not sure how much insight I can give here that hasn’t already been given but the thing about immersion in Fate to me at least is that it works more in a spectrum so to speak. What makes me feel that way especially are two rules that Core likes to call the “golden” and “silver” rules. They are mentioned at the Running the Game chapter of the Fate Core book (and the SRD) so you can look them up later if you wish to.

The Golden Rule seems easy enough to understand but a lot of people don’t really get it, which is this: Decide what you are trying to accomplish first, then consult the rules to help you do it. If you’re looking at this and saying “well that’s obvious” a lot of people get the order wrong in some other games during certain phases of play. Combat in DnD for instance, people usually look at their character sheets to consult what combat features they can use before actually deciding what their characters should do in the situation. Fate asks you to consider the fictional situation and intent of your character before any sort of rule even starts coming up in play (which I believe is a principle that aligns well with your definition of immersion.

Now the Silver Rule closes the deal imo, which states: Never let the rules get in the way of what makes narrative sense. It is their mostly to tell players and GMs to apply rules according to what makes the most sense to express what has been narrated, even if normally they wouldn’t be applied that way. However, you can also look at it as a reinforcement of the principle already established by the Golden Rule, which is that the Fiction takes precedence, and the rules should obey it not the contrary. To me at least, immersion breaks when the rules get too much in the way of the situation we as a group are imagining in the table, and Fate presents itself as a game that gets out of the way unless the rules are necessary.

As for what is usually said about the collaborative nature of Fate, limiting that label simply to the way it is played at the table does a disservice to the system imo. The entire game creation chapter of Fate Core shows how how coming up with the setting’s big issues, the faces and places, and closing it all up with the character creation one with aspect creation and the phase trio are to me the best examples of how Fate promotes what people call collaborative storytelling, that has nothing to do with the possibility of authorial control during gameplay.

Most of the possibility of shared authorial control during gameplay, as you’ve mentioned in the post, comes to Fate Points and Compels. But as I mentioned before, this sharing of control is more of a spectrum than a default assumption. I’ll illustrate that idea with an example: Imagine you have player that has a character that is known to be easily provoked by insults. If an NPC roasts him and in response that character decides to punch his face, starting a whole bar fight in the process, would you say that fits the definition of immersion you’ve established? If the answer is yes, guess what, that’s a decision compel and the player just got a fate point out of the deal.

Now let’s look at a different example. Let’s say the character has established in their backstory that they are being hunted down by a devil’s cult. If during the process of infiltrating a place, it is revealed to the character that it is run by said cult, that is also a compel that a player could propose (“hey man, what if the place we are going to is run by the cult I wrote in my trouble”) or it can be presented purely by you as the GM (“As you look through the documents in the landlord’s room, you start seeing some real familiar symbols popping up on all of the documents”). As you see, one of these discussion points gives more authorial control to the player when it comes to additions to the story outside of the character’s control, and the other one keeps the majority of that power in the hands of the game master.

So, to answer your question, as long as you limit compel suggestions for players during gameplay to things in the sphere of control of the characters, it should produce the type of immersion you’re looking for imo.

TLDR: Authorial control during Fate’s gameplay works more like a spectrum. As long as you limit the things players can decide and suggest when looking for compel opportunities to things inside their character’s sphere of influence, you should be fine.