I would say this is bad practice. You've just had the players roll alertness when they should be rolling empathy. Say one of your characters has Empathy as a Superb skill. They get a huge bonus on the roll. Say they also have only an Average Alertness. You've just denied them a +4 on that roll and they don't even know it! (If they find out they will be rightly angry.)

If you want to test a skill without the players knowing you can always roll for them in secret. That way they have no idea what you are checking against merely that you are checking for something.

However in a system like FATE where players can spend FATE points to modify rolls, etc I wouldn't even do that. I'd have them roll empathy, you don't necessarily have to explain the roll but that lets them leverage their stunts, powers, and aspects as per regular play. If the players figure out what's going on but fail the rolls then their characters are still in the dark. If the players have their characters act on player knowledge you can gently remind them that their character is blissfully unaware of the problem. It can make some interesting tension in the game the same way you don't want a person to open that door in a horror movie! Roleplaying is partly about being able to separate player knowledge and character knowledge.

Retcon the session, just this once

I empathise with your dilemma: the session itself is already immersion-damaging, but so would be retconning it. You can't win either way, so what's the path of least damage? Given the details you wrote, putting myself in your shoes as an immersion-centric player and GM, I would absolutely call for a one-time, exceptional-circumstances do-over.

The balance of issues

First, here are the elements of your question that I'm considering, pro and con, and how they're relevant:

1. The player was extraordinarily off their game for exceptional reasons. Doesn't matter why—drunk, bereaved, recently traumatised—something happened that made out-of-game events entirely eclipse the continuity of the story you're exploring. As you say, the out-of-game situation was such that the session shouldn't have even happened, if you'd known.
2. There wasn't much plot development outside of the uncharacteristic events. If you had a lot of other threads develop in the session then you'd have a knot to untangle if you decided to retcon, but you don't. That makes the situation much simpler.
3. Your immersion is damaged by letting the session stand. There are lots of ways to weave the events into a coherent narrative using reveals of previously-unknown information. That's a considerable amount of work, and carries no guarantee that the immersion damage will be healed. After all, that stuff is being "written in" to explain something that you know happened for out-of-game reasons. Post facto justifications are of the same class as a retcon, actually, when it comes to their relationship to immersion and investment in story—both are editing the game's reality for out-of-game reasons and convenience.
4. It's damaging to your players' immersion. This is the one point where I'm making an intuitive leap off the details you give, but here's why: You are running a certain kind of game, where the continuity and integrity of the story as an unfolding story is important, and that seems to be, from how you wrote about it in the question, a fairly integral part of your GMing style. If I can assume for the moment that this is an established group that has reached the stage of playing well together, your players have adjusted to and entwined their own playing styles and methods with your GMing style. If so, then they've come to trust the integrity of the story in a similar way, and letting this session stand will damage their immersion too. It may also damage their ability to trust the cause-and-effect relationship of things within the world, if out-of-game circumstances are allowed to eclipse in-world causes of events in determining what happens.
5. Retconning is damaging to investment in the integrity of the story. If you retcon now, what's to say you won't retcon again later? You open the door to further retcons for more frivolous reasons, both in your mind and for your players. If you're going to retcon, you need a solid rationale that explains why this time it's justified, but not next time someone is slightly unhappy with how a session advances the action.

Those all, except for (5), overwhelmingly shout "retcon!" Since I believe (5) can be mitigated with a solid rationale, this interpretation of the details of your question heavily tips the balance toward retconning the session.

Rationale

• The session's events were overwhelmingly driven by out-of-game circumstances, that—if you'd known about them—would have justified not playing in the first place.

In-world causes of events were drowned out almost entirely by out-of-game causes of events. Effectively, this session wasn't even about the stuff in your game. That's not going to happen very often—possibly never again—and sets a very high bar for ever considering a retcon in the future. That should give you a solid bulwark against the temptation to start on a slippery slope of retconning for lesser reasons.

So, with that rationale to justify this retcon, but not lesser temptations to retcon in the future, retcon the session. Wind back the clock, cut out the piece of timeline irreparably damaged by an invasion of causality from a foreign reality, and settle back into the true-to-itself reality that the game had already established before this unfortunate series of events.