Just to add to WAAARGHBobo's thoughts:
I think changing the way a game plays overall to suit a plot doesn't make a lot of sense in the vast majority of cases, that's pretty much a given. However, there's a separate question about when to involve a Narrative writer in the process, as writing a plot is just one aspect of the role that we can play in game development (we are all developers, after all).
There's an ongoing occurrence in the game industry known as the Narrative Paramedic issue, (Rhianna Pratchett called this out and speaks on this matter much more eloquently than I) where Narrative writers are brought in near the end of a project to write words to put in the mouths of characters to make them, and the world of the game, make sense.
To give you a hypothetical example of how that can be an issue, imagine you are building a game. Around the table of developers you have game designers figuring out what the world of the game will be. The designers (as happens and is still espoused by some) figure out not just the gameplay on their own, they also figure out what the main themes of the world will be, what the specific themes and unique aspects of each society or area will be, who the main characters will be, etc. and decide they have something very cool. They bring in artists who then make it look cool and move beautifully.
They then turn to the fellow who has been sitting patiently in the corner and say "Hey, George R.R. Martin, write barks for our characters would you? There's a dear chap."
It's a clumsy example, I know, it's meant to outline a seemingly obvious proposition (if you like George R.R. Martin, anyway): If you have someone whose craft is thematic cohesion, world building, character development and so on. why wouldn't you have that person in early to work on those elements of the game? They are aspects that are sublimated through every piece of a game experience, whether you see it on the first read or not.
Working as a team in collaboration, the designer, the artist, the narrative writer and beyond, can reinforce each element of the whole and make each element and team member better. The experience of the game is paramount, but not utilizing the craftspeople you have at your disposal until after you've made choices that will a) likely be not as good as if you'd brought them in to start with and b) inhibit what they're able to do when you do bring them in, would seem to be sub-optimal.
If each discipline has strengths that they can bring to the table, why not invite them?
The 'slows us down' argument is one that I understand. We have have found, however, that the time saved by moving forward without that thematic cohesion baked in from the beginning means that you lose time due to creative misalignment later in production when it's much more painful to make changes. Or you just have to make do with what you have and send for the Narrative Paramedic. Again, that just doesn't seem to be ideal to me. And though I admit to being a bit of an idealist, ideals are useless if they're not practical, if they don't work in reality.
We just find that they do.
It's why we work the way we do on League. We had exactly the same problems with late stage involvement or poor communication and collaboration between different disciplines, which caused thematically discordant champions either through contradictory motivations, inconsistent themes across VO, art, gameplay and so on. We're not crowing that we've solved everything, not at all, but we do believe that working together the way we do makes us all strive to be better at what we do.
That's not to say we nail it 100% of the time, but we've learned what good looks like by starting with how we work together on teams (thanks to dedicated folks like WAARGHBobo, in fact, he wrote the first story sketch to be used in champion production) and what difference it can make when we work as partners with the same goal.
If I'm writing a comic, I'll be very keen to hear what the artists and editors have to say. They have ideas, understanding and perspectives that pertain to their field that will likely be better than mine, or at least spark better ideas between us. Also, the book will include their craft and so I want to make sure the very best of their craft is included as it will undoubtedly make for a better book. To do that I need to ensure we have a working relationship and environment that allows for us all to be at out best.
We believe the same is true for gaming. Understanding what a writer can bring to your game is the first step. Bringing in a Narrative Writer doesn't mean you have to curtail your game to bend to the need of plot. Games are much more flexible than that and so the developers must be to utilize the medium's full range of possibilities, as well as the full capabilities of their team of developers, including designers, artists, writers, engineers and many others who are vital to help something as hard to do as a game reach its full potential.
A development manager asked me the other day about the core motivation of a champion that's currently in development as she didn't feel it was clearly expressed in some concept work that we'd been reviewing.
My answer didn't matter as much as the question. The fact it was asked at all makes me feel we're heading in the right direction.