I feel like a very large proportion of objections to narrative mechanics boil down to an unreflective dislike of the idea that anyone but the DM made that get to narrate anything. D&D has a lot of narrative stuff going on, but it's mostly the DM narrating and some people get very uncomfortable when they think about players doing that, even though realistically we've been inviting players to describe stuff for 30+ years in many cases. You can see this in posts sometimes quite clearly, where people express fears about players ruining the tone or aesthetic of "their" game, sometimes people even threaten to quit as DMs were such a thing to occur!
It's weird to me because in decades of playing games where players heavily describe stuff (particularly from Feng Shui onwards in the 1990s) I've not seen that to be a real problem. I have seen players destroy tone and so on, but that is almost always via obnoxious quoting (looking at you, Monty Python), constant in-character silly business, relentless bad jokes or the like. Whereas offhand I can't think of a time a player narrated something that didn't fit - on the direct contrary, some great ideas and elements have come from there.
But I do think fear of a loss of control is the main issue here, and that it's mostly forever DMs who are uncomfortable. The few players I've seen be uncomfortable have been ones who are either shy generally or who didn't have faith in their own abilities - ironically they're often pretty good at narrating.
The smaller fear I think is that the game will drag if people narrate everything but that surely applies to pure DM narration too (which is rarely seen as "narrative"), and again, that's almost never the case in my experience. What I have seen cause a drag is excessive reading from text boxes and the like, but that's the opposite of this!