My thoughts are there isn't a problem with D&D being a game. There are problems when the fact that it is a game causes rules that don't match other's priorities.
For instance, while there's a lot of alternate Initiative methods talk, there's zero "everyone should yell what they are doing at once, we don't need an order because everyone is really going at once". Everyone can accept that bit as game.
But when you get to rules that prioritize D&D working well as a game but are at odds with other priorities those around the table have. Simulation, realism, narrative as common ones. Ah, there's the rub.
Let me give an example. There's been a lot written about the length of the adventure day, providing sufficient challenges, and also maintainign the balance between the long rest recovery classes like casters, the at-will classes like rogue, and the short-rest recovery classes like monk. The choice given of matching up with rests and sleep is picking a solution that meets the narrative, but doesn't work well across all DMs and all tables in terms of Game. 13th Age, a d20 game, has the simple solution of "a full heal-up (long rest equivalent) every 4 encounters". From treating D&D like a game only, this is 100% a better solution when across all DMs and the varied adventures they run. But it's also a very rejected mechanic by lots of tables, as the negative to the narrative (and to some degree the simulation) is, for them, more than the positive of the net increase in how that works vs. the current.
Again, it's not a problem D&D is a game, and players will gladly accept that. The problem is when the play-as-game come into opposition of other aspects of play that the person weighs more heavily.