I think both because of the coupled, convoluted complexity of much game logic, and because there are fewer open source examples of novel game code available to train on, they may struggle to be as useful.
I know nothing about making a game. I am sure LLMs could help me try to make a game but surely they would help someone who has tried to make a game before more. On the other hand, the expert game developer is probably not helped as much either by the LLM as the person in the middle.
Scale that to basically all subjects. Then we get different opinions on the value of LLMs.
Additionally the problems of custom engines and game specific patterns.
That being said there's parts of games with boilerplate code like any application. In a past game as I was finishing it up some of this AI stuff was first becoming useable and I experimented with generating some boilerplate classes with high level descriptions of what I wanted and it did a pretty decent job.
I think some of the most significant productivity gains for games is going to be less on the code side and more in the technical art space.