Yes that is exactly what a SUMMONER is without a SUMMON. Another mage waiting to be picked off. Taking the summon out of summoner means no class. I don't see a problem here.
They "mixed in" different jobs because reasons they don't want to explain. Why is it just summoner that has to be mixed? The best guess is traditionally Summoner has been either a black mage or white mage that can also call upon summons. The only time this has changed is in the few times summons were disconnected from classes entirely and anyone could use them (like in VII with Materia)
No the "summoner legacy" isn't flash spells. Glamouring the egis into their real respective forms would be "flashy spells" Summoning is about calling in powerful spirit entities to give out big boons. It always has been. They have NEVER been a potato that auto attacks and maybe does a side ability here and there. And in game lore that tries to justify this is completely irrelevant. Its called Lampshading.
And yes they ARE inconsequential. Being shoehorned into a rotation does not change that. You can use garuda, ifrit, or chicken nugget. the result is the same. you press egi asasult 1, and then 2 (or not if you have potato shield) and thats it. ifrit is reduced to nothing more than an (laughably weak) auto attack. You could delete them and just cast assault yourself for the same result. They don't fight alongside you, the auto attack and are can easily be ignored. Before they became immune to damage it was VERY common to find out they died at some point and needed resummoned, not because anything about your rotation changed, just that they wern't there and that enkindle button stopped working. Given todays setting, you would only figure out they died because they're needed to transition into Demis. Their presence on the battlefield changed nothing.
Egis should be removed. If you want to keep Summoner as a pet class sure. We can make that work. But it should be akin to how Yuna uses hers in 10. You summon one, and now you have control of the beast for your own means. Replace your own cast bar and use theirs casting fireball or ice, or whatever summon is used. Pheonix already does this to a lesser extent. I see hardly any complaints on how Phoenix works. It seems universally liked by comparison. If anything Bahamut should get the exact same treatment.
Egis are not summons. they are pets.