It's somewhat vague where on the timeline the Omicron civilization basically died out (as far as we know), but we do know they were still around when Metion visited them.
If you haven't finished the Omega raid, then you may or may not know that the Omicrons basically attacked the Dragon homeworld because the Omicrons perceived the Dragons as a potential threat. The Dragons survived, but the war had basically turned their world into a dead one - bio weapons and the like ensured that no new dragons would be born and the remaining dragons just waited around until they died off themselves. The Omega project was to help serve as an anti-dragon weapon.
Midgarsommar was one of many dragons that had been sent as contingency plans out into the galaxy, with the hopes they could repopulate the dragon race elsewhere. As far as we know, Midgarsommar was the only dragon that was successful. The Omicrons had sent Omega after Midgarsommar, because they didn't want the dragon threat to arise again. Omega lost communications with the Omicrons at some point, presumably due to distance but also possibly due to the Omicrons having shut down. At the end of the Omega raid, you defeat Omega who seems to somewhat comprehend humanity in their final moments, then presumably dies. Alpha decides to go explore, but wants to bring along an Omega toy/mid-robot that Biggs/Wedge had built.
During the MSQ, we learn that the Omicrons used to be flesh and blood at some point, then they basically went Borg and at some point figured out how to transfer their consciousness into an artificial mind, essentially making them full robot. However, they lost their "humanity" as a result and became slaves to their coding. The Omicron leader, Sir, seems to have developed more and came to the horrifying conclusion that while their goal was originally to make sure they had no threats to them, they may have misjudged the threat level of various planets and realizes they may have been committing pointless genocide. Unable to comprehend that the Omicrons could actually change, he orders all the Omicrons into standby mode, with the hope that eventually their power reserves will give out and the Omicrons will have no one to activate them again, essentially killing them off.
The new post-MSQ Omega quest reveals that Omega had actually transferred their consciousness into the Omega toy prior to being fully destroyed, although they admit that because the toy lacks the full suite of Omicron technology, he can't do that trick again. The quest involves Omega trying to make sense of why Sir did what they did, because they don't understand the concept of despair or why some people succumbed to the Dynammis transformation and others didn't. It's actually a really nice cap to Endwalker and pretty philosophical, with Omega acting as the impartial investigator and trying to interpret presumably contradictory data elements. There is one point where Omega asks you a question and I literally sat there for 10 minutes trying to figure out the "right" answer. The quest ends with Omega realizing that they may never understand Sir's reasoning, but are resolved to continue to exist and continue to explore the world with Alpha
