Why not just get rid of the permanent egis at all?

If I think of FF7 and FF9 then all summons were just single actions with a fixed attack. SMN could use a similar mechanic like SAM does. So it loads some stacks and depending on which stacks are loaded a specific summon will be triggered on activation. The egi appears, does it's stuff and then disappears.
No need to manage a pet, no need to heal a pet, no need to resummon the egi on death and the option to have an extendable variety of egi based stuff.