Yes to both questions. You can tell that Ambassador Spock comes from a time further along than we last saw him because, ah, he’s significantly older (on purpose, even!) than…when we last saw him.
The Temporal Cold War advanced the timeline, along with stuff we saw in Voyager (Captain Braxton), to a point where time travel has become so routine that it needs to be policed. There could almost be an argument for a whole series based on time travel like that. But then it would sort of become Doctor Who.
I can’t agree that Discovery seems like corporate nonsense, because it started out by recruiting Bryan Fuller. Fuller was associated with Star Trek before he began creating critically-loved but short-lived cult series like Pushing Daisies. Corporate nonsense is pretty much the opposite of recruiting Bryan Fuller. Happily, he became in-demand around this time, but unhappily it meant he had to leave Discovery. Any fan who seriously questions whether there’s real integrity in this project has completely forgotten or has no idea about the significance of this.
They also recruited Nicholas Meyer, who’s the Lawrence Kasdan of Star Trek. That ought to mean something, too. Apparently does not.