Take a Souls game for example, this is an absolutely amazing case of a game marrying game play and story telling in a really masterful way. When it comes to these games there's usually a scene at the start and a scene at the end and everything in the middle is straight game play. Very rarely will a Souls game interrupt play to show you a thing and when it does it's usually just creatures flying you off to the next area or something like that. That doesn't mean however that the Souls games are devoid of story, the world is teeming with all sorts of details that tell you a story by showing you what's going on rather than telling you about it. Instead of having a bombastic cutscene of monsters killing people, it has you run through the aftermath and letting you put the pieces together yourself. For more detailed parts of the plot you have to look to item descriptions for the things you find in the world which is a sort of interesting take on the tired cliche of data logs.
Another great example of a game that does this is Fortnite. You wouldn't expect there to be much of a plot in a battle royale like that and there isn't but the game does subtle things that show the player how the world is going to change. While other games may make announcements on a news page like "NEW FEATURE!" Or "map changes coming for the next season, Fortnite drops subtle world changes that hint to how the island will change for the next season. For example right before season 4 started a meteor appeared in the sky and a few hints were left around the map as to where it was going to strike. It was the devs way of saying "we are reworking an area" using the game itself rather than just announcing it on an update page.
There's nothing wrong with the classic cutscene, I would never want to see it go. However when a game does marry gameplay and story the experience becomes much much more memorable
