To be specific - around 5:24-30, where he says: "My problem with calling Alyx a great female character in games is not her character itself but her role in facilitating Gordon Freeman as a masturbatory aid for the player."
Because it isn't that Alyx is a 'bad' character, or even a 'flat' character - it is that she is exists purely in relation to Gordon. Which is true of every character in that universe, which orbits around Gordon Freeman's miraculously perfect Van Dyke (twenty years in limbo, not a single chin-hair out of place). But Alyx's role - and sort of her whole persona - is kinda that of a 'combat girlfriend' - she exists only as a female counterpart to Gordon, and ultimately becomes a cliffhanger-bait MacGuffin by the end of Episode 2, dropping the entirety of her character's presence in order to fill out the prototypical 'Damsel In Distress' for the never-to-be-released Half-Life 3.
Yes yes, just made an argument about a game that doesn't exist outside of Valvespace - but 'captured by aliens' doesn't leave much to the imagination. I feel justified in the conclusion given the current evidence. Moving on.
But the devolution from sidekick to mission goal isn't really the problem - we expect that in most action-oriented narratives. The problem is that Alyx Vance has no story arc. Her role is tightly regulated to companion throughout Half-Life 2. She is fully fleshed out, yes, and a fine example of how a real woman might act or behave (leaving aside that the stressess of the post-apocalypse seem to leave her psychologically unscathed - which is a valid argument but one that extends to every named character you meet with the exception of Breen and Father Grigori and possibly Mossman). But she has no character development. She doesn't really undergo any transformative action, she doesn't have a Hero's Journey of her own, she's just tagging along for the ride with Gordon. Gordon's presence - and her sidekick status - lessen her role in the narrative, which is why it's hardly surprising that she turns into another princess to be rescued.
To present an example of a similar dynamic worked in a different way: Chrono and Lucca from Chrono Trigger.
The trial-and-prison sequence early in the game gives you the Damsel-in-distress worked in reverse, with Crono awaiting his death sentence, and the choice to either face it quietly or bust out of jail. No matter which you choose, Lucca rides to the rescue, literally guns blazing, and the two of you proceed to kick ass so hard you tear a hole in spacetime. (I'm paraphrasing, of course). As the game moves on, you see Lucca's reactions to the events around you, and you learn more about her history - why she loves - and possibly hates - machines so much, why she's estranged from (and reconciles with) her father, what happened to her mother and what it meant to her. You see how she reacts to these things and how they change her.
But Alyx doesn't change, really - she isn't given an opportunity to change. She's just kinda there. She kicks ass while she's there, and I'm not taking anything away from that. She's a perfectly normal, believable (if somehow detached) individual, with individual mannerisms and motivations to the extent that her role lets her have them. I can't take away from that, either. But her role restricts her severely, and it prevents her from having much character growth at all, and that is the real problem. And it's something I likely never would have noticed if I hadn't watched the video.
EDIT: Incidentally, Gordon doesn't really have a story arc, either - he's really just following events around. This is an easy pitfall for a character like Gordon, who exists as a player-surrogate (which, if you remember from the video, is the essence of the Mary Sue). It's a criticism of this kind of character in general - but, other games handle this problem by giving us a measure of progression that we can attribute to the character to give some context to all the scenery we're passing; in Chrono Trigger, it was the stories and personalities of Crono's companions, in Super Metroid it was the steady increase of available skills and firepower, and so on. Also, note that in Super Metroid the story was reinforced at the climactic Mother Brain battle by stripping Samus of all her weaponry, leaving her utterly defenseless, which made the baby metroid's sacrifice all the more poignant and the resulting hyper-beam vengeance so viscerally satisfying. A much different feeling from giving Gordon a sniper crossbow and a slapdash dune-buggy ride every few levels.
Supplanting the normal character progression one sees in a narrative with another form of progression readily tangible to the player, a character that exists purely as audience-surrogate can be seen as an active participant in the story, rather than a figure moving through it. And here, Alyx represents a bit of wasted potential - the static nature of her character progression indicates wasted plot potential on Valve's part.