And if the first photon was in some weird quantum state, where it can’t be said to have struck the atoms or not, the second photon will be both extra-delayed and not extra-delayed at the same time.

Maybe it's just me, but these always come out sounding massively hand-wavy and scare off a lot of people (or make you look like an idiot, depending on how much respect they have for physicists). I think the fix would be mostly a matter of coming up with terminology that doesn't fall flat on its ass, but still remains accessible, because, on the other hand, speaking of amplitude distributions would kind of alienate a lot of people.

Anyway, much more importantly - it's a clever little engineering trick to entangle the two photons. However, and I will say I have not researched the topic of quantum computing - doesn't operating them require interference between the different branches of the system at some point to get meaningful results? If so, you would need to get two or more branches to end up in exactly the same state, and nothing in the article mentions that the cesium atoms that have been altered will reset without outside influence.

Anyway, I put the conditional probability that I know everything about the system or quantum mechanics awfully close to 0, so it's more of a curiosity (how would you make it work?) instead of a genuine concern.