There are a few key expansion on the original idea:
Sephiroth wants to transcend "One Winged Angel"
Sephiroth knows that if he uses Meteor in infinitely many worlds, Holy is also cast in infinitely many others. That makes him, in the grandest of schemes, half-complete. His whole motivation is driven by his inherent antagonism, being a demigod, so he needs a superordinate scheme to win across the board. This involves elevating Cloud to the position of a Christ-like figure, then toppling him to pull the rug out from under humanity across all worlds.
Cloud is half-dead, like the player
Make no mistake, I'm not saying this is meta. However, the theme is not arbitrary because it echoes the change between the OG player in 1997 and whoever he/she has become today. The intervening years have made the player stronger and wiser; Cloud is now able to hop between worlds. The intervening years have made the player older; Cloud is now more profoundly plagued by SOLDIER degradation, which is a form of half-death, like ageing.
These are the two main narrative threads I think are holding together NU FF7.
How did this reality get started even?
Good question. A cynical, but totally legitimate answer is that the devs decided to cash in on nostalgia. However, I think the original ending of FF7 is intriguing enough to lead into the Remake series.
Red as The Watcher
If Red has always been in-tune with the deeper reverberations of the lifestream, it could be that he's the one looking into these realities.
Of course, this is contradicted by the focus on Zack, which suggests the remakes are using the compilation series as a framing device. I wouldn't be so quick to make this judgement, though, since the overarching and central themes are all expansions of what was in the original.