You are mistaken about the basics of the time travel alteration, and from there reaching the wrong conclusion about thinking "Meteion only becomes the Endsinger later".
Multiple incidents from the beginning of Endwalker indicate that the world is already in a state affected by our trip to Elpis. If there ever theoretically was a timeline where the changes were yet to be applied, we are not in it, but more likely there never was such a timeline and it has always played out this way.
- At the beginning of the game when Hydaelyn speaks to us on the ship, she is already aware that we are approaching the point where the time loop resolves.
- Argos is willing to let us ride him when we first (to us) meet him on the moon because to him, we already met him in Elpis and earned his trust there. This is specified in the mount description as a quirk of time travel.
- As you note, the Endsinger – the collective Meteia as opposed to a single Meteion – already exists.
Also, even if there theoretically was a "pre-altered timeline" without our interference at Elpis, I don't agree that it could change the Endsinger from a pre-existing entity with its own history to being connected up to the Meteia.
And I don't believe there is any need for a pre-altered timeline to exist, particularly in this universe's time-travel rules. The ultimate example of this is when we travel back in time during the A12 fight to save our earlier self from Alexander's laser beam. There cannot be a pre-interference version of this timeline, because then we would be dead and unavailable to be sent back. Alternately, if something else saved us the first time, then there is no need to overwrite the event. Therefore logically no version of events occurred except the one we experienced, time travel and all.