Under the NarraChara perspective, when does this kick in? When the game proper begins, the human is set in the Ruins' flower patch, conscious. Later in the game, though, while sleeping or otherwise unconscious, we are given dialogues which wouldn't be from Frisk; these are assumed to be from Chara. So, could Chara be delivering the opening movie and, "Name the fallen human." ("The true name.")? Especially if the latter, this could support Chara being "the one in control," even of playing at all!
There are a few trains of thought in regards to when NarraChara kicks in.
First, you are very right that Frisk falling asleep or otherwise falling unconscious seems to yield memories that belong to Chara! This is one of the supporting pillars of the theory, so the Chara theory must be in full effect by this time.
There are two likely points when this could have happened, and they are very near each other. The first is the naming screen, where the player is instructed to “name the fallen human.” It is revealed at the end of the true pacifist route that the player is naming Chara here. Chara describes themself in the genocide route as:
This makes it very possible that the character naming screen is the point of Chara’s awakening.
The other likely point takes place shortly thereafter, when the human talks to Flowey for the first time. The reason for this is that the human is initially unable to check the flowers and get narrative text in the first room, but is able to do so after talking to Flowey. Perhaps hearing Flowey and Toriel caught Chara’s attention? Hard to say. Either way, it’s known that Chara is awake under the NarraChara theory when Frisk is able to check things and receive descriptions of them. It’s also known that Chara is in control of a genocide route to some capacity when the red text descriptions are available in Toriel’s house. The time of Chara’s awakening is narrowed to a small window of time in the beginning of the game, which is pretty good!
So, is it possible that Chara is the one narrating the opening sequence?
On the one hand, it would be easy to dismiss this with an easy “no” answer by saying that the opening sequence is asynchronous with the events therein, but perhaps that would be oversimplifying the question. After all, the sepia memory sequence during the battle against Asriel at the end of a true pacifist route is implied to be Chara’s memories showing through – at least, under the NarraChara theory. Perhaps one could extrapolate that the opening sequence is a series of Chara’s memories as well. Indeed, many of the screenshots of Chara walking up the mountain likely are. However, the intro also includes historical pictures of humans fighting monsters, and these are implied to have taken place a great deal of time before Chara was born, judging by the weaponry used in the pictures. Those pictures are less likely to be such.
There’s also the issue of whether or not Chara is awake at this point in the story, if it is synchronous. Chara addresses this in the genocide route ending:
What is referred to by “your power” is ambiguous. Is it the act of naming the fallen human? Is it the simple act of booting up the game? Hard to be sure. Whether the theory holds any water is up to one’s interpretation of this line.