Do computer scientists care about the complexity of specific games still?
(Expanding my comment.)

I think your assessment is correct -- it's much harder to publish results of the form "game X is NP-hard" today than it was last century -- although it is not something I've looked into.

I think the reason for this is the following. 

The first few examples of NP-hard games are interesting. It means we can say "many games turn out to be NP-hard". That helps motivate research on NP-hardness and helps scientific communication with the general public.

However, once, say, half a dozen examples have been established, what is the value of adding more examples? Of course, there is value in having more examples, but it becomes more marginal. Each successive example is less surprising and doesn't really change the above statement.

Moreover, many of these proofs are quite messy -- not the sort of thing a reviewer is keen to work through.

Note that this phenomenon is not exclusive to results about games. I think there are many papers that struggle to get published today even though they'd sail in 30 years ago. For example, Karp's 21 NP-complete problems is a celebrated paper, but a similar list of NP-complete problems would struggle to get published today. The reason is simply that those 21 examples were surprising then, but are not today.

An important side note: In essentially all examples, the original game is not NP-hard because it has a finite, albeit extremely large, state space. Thus the NP-hardness is proved about some generalization of the game. Such as Chess on n-by-n boards with slightly modified rules.