Something like a series of matches with certain opponents should feel special whereas with current WWE it is the norm.
For example, Benoit and Booker T had a best of seven that culminated in a shot at the WCW TV title. It was not the norm for WCW. AEW has done the Labors of Jericho to get him a rematch with MJF which was a fresh spin on that concept. You also have had people running through Dark Order to get to Page or the Elite to get to Omega.
In the later examples, I think that might help with the randomness because when you feud with the "leader" of a faction, you get the stable hands before the big hoss. Lets everyone have the shine. That's a very WWF/WCW/NWA thing (the Horseman, Freebirds, the Flock, the Nation, DX, Ministry, Heenan Family, Dungeon of Doom, etc.) to feed a faction to a star.
Do you watch Dark or Elevation? They tend to be very random at times. I can get behind that critique. While not perfect, I wish AEW would develop some more under card feuds. Not just Mid and upper, which to be fair, is 100% better than WWE who really only tell Main Event stories anymore.