One avenue of thought that people rarely consider is the Corypheus and his Lyrium Dragon. That could have implications for the elven gods, and the relationship between the Old Gods and the Elven Gods and their separation.
Suppose- just as a hypothetical- the Old Gods were spirit-companion-bond-partner-things for the Elven Gods? That the Old Gods were a connected but separate part of the Elven Gods- something like Hakkon, but bonded like the Lyrium Dragon to Corypheus?
There are a couple different elements in DAI we could use as a basis for consideration-
-There's a reoccuring theme that Tevinter magic is really elven magic reclaimed- not just Solas's orb, but magical techniques- but what's really relevant is Corypheus's survival- and how Solas reacts. Solas admits in Tresspasser that he had not expected Corypheus to learn the secret of effective immortality- which could suggest he himself knew what that secret was- and we learned from DAI that Corypheus's survival rests as much in his arch-demon as in the Blight. Were dragon-bond-pylacteries something the elven gods had?
-In the post-Adamant Solas and Cassandra companion dialogue, they talk on the nature of Corypheus's ArchDemon. That it was manufactured, yes- but Cassandra also gives a musing on wondering if that's all that the Old Gods were? Pets to beings no longer present? Solas plays coy that it's not quite like that- but considering how much DAI uses companion banter for foreshadowing exposition, and who's involved, it's likely an important lore-tease.
-Jaws of Hakkon gives an avenue towards spirit-empowered dragons- which seem relevant when discussing the ArchDemons. Dragons that are more than just dragons, beings approaching godhood but may well be artificial, like the elven gods.
-The unique sort of role and relationships that the ancient elves had with spirits- where spirits like the Forbidden Ones were considered more or less the same as residents as the elves, while Cole suggests the sort of being that hovers the balance between distinct races.
Put it all together in a rough theory, and-
-Elven Gods, aside from being shape shifters as Dragons themselves, also had dragon-phylacteries as part of their empowerment.
-Old Gods were those dragon-phylacteries, part of the elven gods' 'effective immortality' in which the Elven Gods put themselves into. Since the dragons were extensions of the god-mages, and dragonhood was proof of the divinity that the elven gods could use, the elves remember the god and not the spirit-beast sidekick. The Old God cults are more from the
-Old Gods are dragons in the same way Hakkon is a dragon- spirit-possession of a special sort on special beasts, which the Ancient Elves (masters of magic, friends with spirits, creater of monsters) would have been able to do.
-When Solas raised the veil, he did so in a way that separated the elven gods from their dragon-phylacteries. Elven gods- or at least their minds- trapped in the fade. Dragon-phylactery Old Gods, trapped in their prisons. This helps as the origin of the 'Dread Wolf trick' myth the Dalish have- even as the Old Gods gradually got confused for the Forgotten Ones and/or the Titans in the myth evolution's mess.
-As dreamers/through their Old Gods bonds, the Elven Gods reach out to the ancient tevinter magisters. They lead the Magisters to the Golden City, which has power/is a key to freeing themselves, and possibly pursuing revenge.
-The plan backfires. Magisters get blighted. The Golden City darkens. This is the point that the Old Gods truly fall silent- whereas before, Corypheus had communed with Dumat even as the religion (and Tevinter) seemed to decline, afterwards no one seems to.
-The corrupted Old Gods phylacteries, when infected by the Blight, become the Archdemons.
Or something like that. There's a lot that's unknown- but it can fit with a number of other theories as well. Even the nature of Flemeth's motivation- Flemeth saving the Archdemons souls, but turning them to purposes the Elven Gods would not lie- in a 'revenge is living well' sort of path of spiting the elven gods by reclaiming/saving their Old God phylactery pets.