But do we even know for certain that Anabel is from the Advanced timeline? Yes, they use the Emerald Frontier Brain theme for facing her in the Battle Tree, but we also know that OR/AS' Battle Resort had plans to be expanded into the Battle Frontier, starting with the Battle Tower. (ignoring how much of a trolling tease this was as a feature we'd never actually get to experience in the game) She could have come from another "Ultimate Weapon universe" where the Frontier was actually completed and an Ultra Wormhole opened up near it and she fell in. (I wouldn't be surprised if they somehow tied Ultra Wormholes occasionally opening to after effects of the firing of the Ultimate Weapon in a later game)I don't think Looker's a faller, in my opinion. I think the coincidental circumstance is just that.
We know that there's two secret bases near the Battle Resort where the International Police staked out a criminal. The theory that Looker was the officer, and was knocked out by the criminal, and was brought to the Resort on the tide.
Then there's the theory that he was the officer on the SS Anne back in the Kanto games, but disguised (which is one of his signature traits). And that he tried to stop Team Rocket but got knocked overboard and floated to the Battle Resort.
Or here's a third theory. After Guzzlord ate the previous faller 10 years ago (yeah, they heavily imply that the last faller was eaten. Or at least killed by Guzzlord.), it could've blasted Looker and Nanu with such force that Looker was rendered unconscious and eventually ended up on the Battle Resort, with mild amnesia that would've gone away by the time his timeline's Sinnoh events rolled around.
I never said the Advanced timeline didn't exist still. Because despite needing an app to do so, it can still connect to the Ultimate timeline. Also, my theory didn't come from Anabel's dialogue at all, just thought of it on my own. If they can invade the Ultimate timeline, as shown in Sun and Moon, and they can travel to the Advanced timeline, as shown by Anabel, then there should be no reason why they can't invade the GB timeline, or the anime or the manga.
And I mentioned that there was no time travel in the Advanced timeline and Pal Park myself. But... I forgot the Celebi event. Which does reinforce my point, that outside of Celebi, time travel is non-existent in the Pokémon core timelines post-Time Capsule.
The GB timeline is the only one that's completely cut off from the other game timelines. (I'm obviously not counting spin-offs, since those are typically cut-off as well, unless you can send Pokémon to/from those games.) And a reasonable reason for why it's cut off from its sister timelines is that it was destroyed.
Then again, one could argue that due to the Red/Green/Blue/Yellow re-releases, that that timeline is still intact, just extremely... basic, compared to its sister timelines.
As for your new theory, yes. Lillie, Lusamine, Guzma and Sun/Moon are all fallers. So if we do actually go back to a "ten years later" Kanto as so many are suspecting, with Lillie as your rival, that would explain how and why the Ultra Beasts (who are now obviously known as Pokémon) would show up in the post-game for collection.
Though... I don't remember reading anything about secret bases the International Police had set up near the Battle Resort...