Poor Terry Matalas. It's clear from numerous post-season interviews that, for as elaborate as S3 became by the end (rebuilding the Enterprise-D! Bringing back Ro and Tuvok! Changelings and Borg and Lore!), his original vision was yet more elaborate. Apparently he originally planned to have Janeway and Kim also appear, and to show Ro still alive in the brig with Tuvok at the end of the season. The man clearly was dreaming big.
Given that, it seems slightly implausible that he would omit material purely out of carelessness. And the absence of Alexander seems like a pretty large omission -- especially in a season that was so focused on the parent-child relationship and the idea of "the next generation". Yes, there are all these memes about Worf forgetting Alexander, but that doesn't strike me as the kind of fan service Matalas was going for.
From a storytelling perspective, omitting Alexander seems pretty similar to why Odo was mentioned adoringly as "a man of honor" but not named: there was already a lot of backstory and reference being woven into the story, and throwing out a random name -- or a random concept like, "Oh yeah, Worf has an estranged son" -- would create too much to unpack.
Likewise, it seems like they wanted Worf to have a paternal presence with Raffi, so omitting Alexander simplified that story.
But still: in a season that was all about parents and their children, it seems significant that they couldn't find any way to reference him.
Unless...
Worf has a memorable scene with Raffi where he tells her, "Don't presume to know what I have sacrificed" (or something to that effect). Surprisingly, that line is never followed up on... explicitly.
But I suggest that that is where we learn of Alexander's fate: Worf has lost his son. Whether to death or desertion or deep undercover work, who can say? But we have an open question -- where is Alexander? -- and we have a vague statement that is never otherwise explained -- that Worf has sacrificed a great deal -- and given how much the rest of the season ties itself together, I suspect this was meant to be a subtle nod to explain away Alexander's absence.
Why not make it explicit? Why doesn't Worf tell anyone about Alexander? I argue it's because they wanted to save the "grieving parent" story for Riker + Troi, especially Riker. Explicitly portraying both Riker and Worf as grieving fathers would create an elephant in the room too big to ignore, and would've taken up much more space in the story.
So, instead, poor Alexander is consigned to a mysterious comment from his father -- perhaps fodder for some future tie-in novel, or perhaps someone we might meet in Star Trek: Legacy.
Are there other theories as to where Alexander might be, or why the writers did not mention him?
The decision to not even drop his name in this season of Picard really hammers how bad of a father Worf really was. This is absolutely something that should be explored in a Worf-centric story. There's a lot of context for how Worf was raised by Humans, but Alexander was not.
DS9 tries very hard to deal with this, but they actually make it worse by mistake.
Worf: "I cannot fix the mistakes I have made, but from now on I will stand with you. I will teach you what you need to know to be a warrior, and you will teach me what I need to know to be a father." Alexander: "Let's see if you mean it."
And clearly he didn't. He was too caught up in his own warrior's journey to consider Alexander. Alexander left, remained Klingon on Klingon ships and continues to be estranged from his father.
Even though I wouldn’t be terribly surprised to learn that this is the case, I think you’re making a lot of assumptions based on little to no evidence. That the notoriously private Worf didn’t mention Alexander to anyone doesn’t necessarily provide that they have a strained relationship; Alexander may very well have died and Worf suffers in silence over it. Conversely, Alexander may be a prominent member of the House of Martok and is well-known in the Federation so no one needs to ask about him.
I’d like to think that Worf’s emotional maturity means that he probably did or does maintain a better relationship with Alexander, and it’s simply that he didn’t come up in the context of what was going on in the story.
I would like to believe that as well, but we are left with very little to make us believe that other than good will towards Worf. It seems like a particularly glaring oversight to have the season of Picard featuring Picard and Beverly's long lost son, Geordi's daughters, memories of Thaddeus Riker's death, and the first time we've seen Worf in a long time not explicitly mention at least once that Worf is also a member of the parent club.
Instead what we do see are exclusively episodes where Worf's relationship with his son is not treated as a core part of Worf's character - even attempts to reconcile go basically no where.
A single line to Enterprise-D crew about how Alexander is the commander of an entire Klingon battle fleet now and that fills Worf with great pride would have solidified that the characters reconciled. Hell, it's kind of weird that no one asks about Alexander since all the Enterprise-D crew readily know that Worf has a son. But instead, just like the writers, Worf has apparently forgotten that he has a child and so is written as though he doesn't have one.
The fact that the oversight is so glaring is what makes me think that we are supposed to read in to the vague line about "sacrifice", and that we are supposed to infer that it's about Alexander. It's subtle and I think it's unsatisfying narratively, but I think that was the intent.
As to why they didn't just make an offhand reference like "Alexander is in command of an entire fleet"... I'm guessing they felt (rightly or wrongly) that it would be weird not to do more with the idea of "Worf as a father", so.... they opted to quietly make him Not A Father (Anymore). Which, to be clear, I'm not saying was a good decision, I'm just trying to do some tea leaves reading here.
This is a good point and perhaps there is unused footage that mentions Alexander in more detail which would understandably be cut out if they weren't willing to lean in on it. It's easier for us to forget Alexander existed and to wonder "maybe he died" than to waste lines on that kind of backstory - at least in some writer's opinion which I could see an argument for.