I'm in the "don't get rid of the egi's" camp, but I do understand the issues that they have (SMN is the dps I play when I'm playing dps). Here's my suggestions:

Egis: egis do their AA as normal. However, the Egi Assault skills are simply abilities they they grant the summoner, not commands to the egis. (Probably doesn't fit, lore-wise, but looking for compromises here.) Hopefully, this will help with the ghosting.

Demi-summons: Treat these more like the MCH's Automation queen. Once summoned, they go through their rotation on their own. Like queen, we can still command an early finishers in case of boss jumps, but hopefully will help remove the "following" issues as well as needing perfect timing.

A demi alternative would be to use the AST "pet", Earthly Star. Put them out at a spot, they stay there, and if the boss moves out of range, too bad -- that's part of the challenge (tbh, I'd probably hate this in practice -- just throwing out options that I know they have the ability to implement).