It's actually stated after Emet's death that he wasn't maintaining Fake Amaurot, he'd just created it with enough aether that it kept standing afterwards. It's going to disappear eventually, but until then, it keeps standing.
(For the obvious MMO game design reason, honestly; Amaurot has to be there indefinitely so we can keep being able to go do quests there, so it has to stay there until the game shuts down. If he were still maintaining it then you just back-date the same problem; eventually someone's gonna kill him, and then he can't maintain Amaurot anymore.)
Also, Hythlodaeus knows Emet's last words because we tell him. There's a cut in the Etched in the Stars cutscene where the WoL explains recent events.
As for 'how your band of misfits survive'... I don't even know what you're talking about, so I'm going to scattershot all the simple explanations for what you could be saying that I can figure.
Why could the Scions go to Fake Amaurot in the first place? Because it's physically there, they can walk there, and Bismarck filled the local area with oxygen.
Why could they go into Dungeon Amaurot? Because Emet-Selch just has a door to that place sitting in Fake Amaurot.
How could they then fight in there? The good old-fashioned ultraviolence.
How could they break into the arena for the Dying Gasp after the fight? Hades had been actively manipulating that one, and at that point was freaking out and got the snot beaten out of him; the barrier keeping them out was weak enough to bust through, if it wasn't already.
How could they find you in Amaurot in 5.3? They... uhm, didn't. Elidibus teleports the WoL and Y'shtola to Fake Amaurot, and after Etched in the Stars, they teleport back out.
How could they find you after the Seat of Sacrifice? They... also didn't. We only reconvene with them after leaving the Crystal Tower; presumably they were holding the line somewhere further down during all that.
Yeah, I think Emet's gonna be a presence in Endwalker, even if I really, sincerely wish he wouldn't be. But it's going to be a posthumous presence, probably in Echo flashbacks or recordings in Mare Lamentorum (or whatever other Ancient structures we find). He's not coming back for some last laugh--and if he does, frankly, I quit.