Lets make this brief; FATEs are bad. Really bad. Worst piece of content imaginable, actually.
Solution? Copy XI, grab your torches and pitchforks folks.
Take a look at this;
You may not like it, but this is what peak NM farming performance looks like. The basics;
You kill trash. Trash either spawns a NM, or drops an item you trade to a specific point to spawn a NM. NM drops item that is used to spawn higher tier NM (for example, Jailer of Temperance drops "1st Virtue", which is used with 2 other drops to spawn Jailer of Hope). Progression like this continues until you reach the "final boss" (Jailer of Love here, Pazuzu for Anemos, for example). I believe this would be a far more engaging mode of gameplay than Eurekas current AFK FATE train nonsense.
You'd enter Eureka, go farm items that are required to spawn NMs. Spawn NMs. Kill NMs, get drops from NMs that lead to fighting bigger NMs. Make it function effectively as Levequests/Treasure Maps, so there is effectively a Claim system for NMs (I spawned Pazuzu, it's my Pazuzu for me and my group to fight, not some laggy worldboss where my individual performance in the fight really doesn't matter since I'm just 1 out of about 100 people wailing on it...). Pop items could be obtained from a NPC at the camp, you kill something, it drops an item you trade in for the pop item (this would allow a single item to require drops from multiple enemies/NMs), using the item effectively triggers a Levequest for you and your friends to fight, probably want to make sure you're in the right area first though.
Issues? Well, I guess too many people could want to fight King Arthro, for example, and you'd have lots of groups stacked up fighting their own version? Well, personally I think the whole progression system fixes that. You wouldn't just go to his spawn area, kill crabs, spawn him and fight, you'd likely need to do some of the lower tier NMs for items to spawn him as well. That would split people up across the map. If that's no enough, you can have it so only 1 group (or just a low number) can fight at a time, don't see that as too much of an issue if you scale how long the fights last correctly, and there's always the option to go farm items to spawn Cassie while you wait for Arthro to clear out. Heck you could even straight up copy maps and just have stuff spawn in random locations, although I do like that each NM has its own area currently, and you'd need to not make that 100% random since spawning Snow Queen where Behemoth spawns, for example, would be ridiculous.
And what FFXI tribute post would be complete without mentioning Absolute Virtue (who is oddly missing from the above NM chart but whatever), or his more infamous brother, Pandemonium "People literally passed out fighting me for over 24 hours" Warden. Anemos is probably going to die a slow death as people don't need to do it anymore? Not a problem! Add a drop to Pazuzu, that when combined with similar items from Snowzuzu and Lavazuzu allow you to spawn a super secret mega ultra boss! Suddenly all the NMs in Anemos (and Pagos once Pyros arrives) have a second wind as we work through the NM progression to spawn all the Zuzu's to spawn and fight such a boss. Bonus points if Light farming is added to Anemos (and it damn well better be in Pyros), given we'd have solid reason to go back there.
OK, that's all. Feel free to downvote the mere mention of XI.