I'm going to run FATEs. I'm going to get my weapon. And I'm going to like it because it isn't a Mythic weapon from FFXI where it took two years of daily raids to get enough Alexandrite to get the fully upgraded equipment.

If you want weapons without anywhere near the foot work involved in Animus quests, go get your Leviathan weapons.

...Did you just say that wasn't an option because they're hidden behind HM and EX primals?

Too. Damn. Bad. If you want to progress, you're either going to need to put in time on Atma, or you can put in time on Primals. Pick where to put your time. This is an MMO, and while the majority of the game's content can be done fairly quickly, there are some things that will require time. The sooner you accept that you WILL need to sink time into getting high level weapons, the sooner you can decide which route you want to take.