@SOC I think you can generally assume that anyone who makes it past the opening menu is, in fact, here to play your game, and doesn't need you to immediately give them Cool Story Info to get them to play! They're already there, so you can choose how fast to show them more than moment-to-moment gameplay narrative
So, here's an example from what I think is pretty commonly agreed to be a game with good narrative storytelling: Undertale. (I will try to omit spoilers, but the game is what, four years old at this point?)
Almost every piece of set dressing in the entire game can be examined for dialogue. This dialogue typically is just goofy, but occasionally portentous... not that the player would know!
There are about three backstories in Undertale. There is the backstory given in the pre-menu picture series, there is the story of the group of friends the character makes in the middle section of the game, and there's the narrative-driving story that underlies the whole plot, not just bits and pieces of it. When is the player first filled in on the narrative-driving story?
The very last area in the game, right before the final scene.
The vast majority of playtime is still spent with a (non replaying) player totally in the dark about this unspoken story. The player and the main character are the only ones who don't know what happened. The world is full of objects that visually reference this story or have tooltips that hint at it. There are cutscenes, that, on replay, seem so blatantly obvious you can't believe you didn't get it the first time. But the player doesn't know, so they don't notice those things were
important until later, they just absorb them as a cute little detail.
This leaves the game with inherent replay value even when no new material is unlocked, which is an awesome structure for short, tightly plotted games, but not so great for long, sprawling ones. However, I think the main takeaway is that if you build your game in such a way that players are encouraged to check every rock for information, they
probably will, and you can quite reliably dispense important background info that way. Save cutscenes for choke points where the plot is being advanced by other characters doing something. Don't show events the main character wouldn't know about, just their consequences. Keep in mind that things that have more consequences should be mentioned more often, and thus be less likely to be missed by a player.
You'd think this approach would lead to a very minimalist game with a lack of plot development, but it can actually make for a game with a plot that seems much more elaborate and lush than if information about what's really happening was easy to get.