This is the nub of the issue here: a story can make you cry by empathising with the protagonist (or another character), but a game (when viewed as a formal system) cannot do this. It follows that the only way that a videogame can make you cry is by using narrative tools that have nothing to do with games as formal systems whatsoever.
Although I think there's something to this, I have several problems with the original argument and also the sequel.
To start, Bateman specifically excludes crying out of frustration, which I feel is both unfair and self-defeating. Research on the fundamental emotional causes of crying is slim, but the idea of frustration or helplessness figures significantly. In a wide-ranging review from 2000, Vingerhoets et al. asserted, "Crying in adults is associated with feelings of helplessness and powerlessness, but it is unclear whether these feelings are components of the appraisals that bring about crying or are simply feelings that accompany crying" (1). This may seem to be applicable only to cases where the emotions involved are negative. In 2003, however, Miceli and Castelfranchi speculated about a general model that placed frustration (or the anticipation of frustration) at the root of all crying experiences (2).
These papers aren't definitive answers, of course, but if taken seriously then Bateman's exclusion, properly applied, could also eliminate all episodes of crying in response to any media, ever. In which case it would be true that, excluding frustration, a game can't make you cry, but it would also be rather uninteresting to say so. In the focus on helplessness, however, we can see the shape of a key point. As Bateman notes in his original post, "when a character dies during gameplay, the option to reload is essentially always present," meaning that the player is never helpless.
For instance, in Mass Effect a certain conversation can lead to the death of the character Wrex. A player, even one who is extremely attached to Wrex, is unlikely to cry in response, because he can simply reset and try the conversation again. Even if he has gotten the game into a state where he cannot save Wrex in the conversation (I believe that this is possible), he always has the option of reloading from a much earlier save or replaying the game completely. Although Wrex's potential death can plausibly be called a part of the game's rules, it's unlikely to inspire tears because the player always has control. The game's capacity to produce feelings of helplessness is thwarted by the player's ability to exit and re-enter the system.
One could in principle adopt a similar approach to films or books by, say, always stopping The Fellowship of the Ring before Boromir's death. However, films and books are fundamentally immutable; part of the sadness of the events in them is due to our inability to change the outcome (frustrating!). If our hypothetical reader ever finishes the book, Boromir will fall, no matter how much he wishes otherwise. Games adopt this immutability through the cutscene; by employing a traditional narrative form the requisite helplessness may be imposed from outside the game's systems.
Is there some way we can get around this?
In Miceli and Castelfranchi's model, crying prompted by positive emotions results from the relief of actual or anticipated frustration. This knowledge could be used in the construction of a game system. For instance, at the end of Paper Mario: The Thousand Year Door, Mario confronts a boss who is extremely difficult to damage and appears to heal herself completely, promising a near-impossible battle. At this point the game pauses for a cutscene in which the positive messages of all the people Mario has helped cause the boss to become vulnerable. Perhaps I'm a big ol' softie, but I got a bit misty-eyed at this point, even though the things Mario's supporters say are all rather generic and silly. Was I moved by the Punies' message of hope, or did the relief of learning that the game would not frustrate me contribute to the response? I don't know the answer, but it's something that bears consideration.
It also strikes me as possible that game systems could be used to inspire a cry all on their own. As I have mentioned, I find the last fight against the shadows in ICO quite moving for reasons that are, in the context of that battle alone, almost wholly narrative. As each of the shadows is defeated, a sarcophagus lights up, implying that the shadow was a boy just like the protagonist, locked in a coffin to die. This has little to do with the mechanics of that fight. Yet, it forces a reinterpretation of the rules that governed the all the preceding parts of the game. The shadows that have been viewed as bitter enemies, good only for striking, are revealed as pathetic slaves of the castle's queen, and one feels almost guilty for fighting them.
By positioning the source of frustration in the past, the reveal of a system could prompt catharsis without losing the feeling of helplessness. It seems to me, therefore, that it should be possible to design a game in which the player partakes in a system before he realizes its awfulness and cries from regret. Has anyone cried from playing Braithwaite's "Train"?
The important thing in designing a game system to inspire crying would be to shift the moment of helplessness away from the moment of catharsis, either by placing it in an avoided, fictional future, or by embedding it in unavoidable mechanics. Using these approaches, games could incorporate the kinds of frustration that cause tears without necessarily relying on dramatic cutscenes.
I also think that Bateman's argument conceives of emotional events far too narrowly. If I read in the paper that Joe Schmoe has died, I'm unlikely to weep. Now, it may be that Mr. Schmoe was one of the kindest people in all of Boston, and everywhere he went cute animals frolicked on the grass and bluebirds sang and the poor were fed. I didn't know the guy, though, so I'm not too cut up about his passing. Yet if someone from my family died I would be sad and cry, even if everyone else found him to be very poor company. When it comes to emotional responses, the stakes matter. As Miceli and Castelfranchi put it:
...if crying is to result, frustration should induce a certain amount and kind of suffering. The degree of suffering of course depends on the value attached to the frustrated goal. The kind of suffering... is strictly linked to the kind of goal at stake...
To bring this back to games, consider Agro's fall in Shadow of the Colossus. If you watch it as a snippet of film, there's not much impact to it. Maybe you feel a little something for the man who was riding, but what happens to the horse won't be too affecting. The hours of play that precede that moment, however, generate an emotional attachment between the player and Agro. All Agro does is come when you call and behave as much like a horse as is practical in a game: Agro is a system. Yet interacting with that system produces emotions that pay off at the moment of the fall. If a player cries in response to this, the responsibility belongs both to the player's helplessness and his affection for Agro. In this sense, the game made him cry.
Bateman attempts to address this line of argument in his original post in this way: "since a purely narrative form would have allowed you to deepen your relationship with the characters without interaction, it's far from clear that allowing interaction is enough to make the claim of a game making you cry..." This is ridiculous, because here he's insisting not that we prove games can make us cry, but that their systems can uniquely make us cry, in a way that would not be possible with any other media. This is a loser's game, since Bateman himself (accurately) claims it's trivially possible for a narrative to inspire emotions. It's true that in some alternate universe, Ueda's classic novel Shadow of the Colossus might have made me love Agro just as much as I did in our universe's game of the same name (obviously very different techniques would have to be used). This counterfactual is completely irrelevant to the question of whether the game, which actually exists, actually inspired affection, much less whether some game could possibly do so.
There is a certain muddiness here between "narrative" components and "systems". Would I have loved Agro as much if he were a lizard? A featherless chicken? A square with four squares sticking out of it? Perhaps I would not have. At the same time the graphical (i.e. narrative) depiction of Agro as a horse serves to contextualize the system he presents and make the game's rules intelligible. At any rate, I'm not sure that we learn much about the player's emotional response by making the distinction. The salient question is how design can be used to inspire emotion. In the context of a game, it is difficult to inspire helplessness without resorting to extreme difficulty or heavy-handed authorism. The solution is not to declare defeat, but to get more creative with design.
1) Vingerhoets AJJM, Cornelius RR, Van Heck GL, and Becht M "Adult crying: A Model and Review of the Literature". Review of General Psychology 4 (2000), p. 354-77. DOI: 10.1037/1089-26220.127.116.114
2) Miceli M and Castelfranchi C "Crying: discussing its basic reasons and uses". New Ideas in Psychology 21 (2003), p. 247-73. DOI: 10.1016/j.newideapsych.2003.09.001