It's less an identity crisis and more of a lack of synergy. We have an identity. We deal damage over time, and micro manage burst that is spread between ourselves (DoTs/Aetherflow), our Egi, and Bahamut. DoTs and Aetherflow are part of this identity, and should not go away just because it doesn't 'feel' like Summoners of the past. We try to contribute just as much as our Summons do, and it's only by working in tandem with them that we see our best results. The problem is very little of the disparate parts of our kit are allowed to interact with one another in a meaningful and fluid manner. In other words, the Egi and Bahamut do not enhance us enough, nor vice versa. The abilities that currently have any direct interaction between the Summoner and a Summon are Sustain, Rouse, Enkindle, Contagion, DWT, Devotion, the Ruin IV/Enkindle proc, Wyrmwave, and Enkindle Bahamut. That's it. Bahamut carries the bulk of the meaningful interaction between any part of our kit now, taking over the role DWT had prior to Stormblood. That's why I believe consolidating Rouse, DWT, and him into one button and removing the Aetherflow lockouts would help out a ton with our flow. It leaves only the egi themselves to worry about, that way.
While I hope they fix the egi to work better, I sure as hell don't want them to work exactly like Bahamut. He has flaws too. He's actually worse than both of them, and hits even harder in order to make it more aggravating. Imagine Garuda chasing after you in-between each of her 1s spell casts when her GCD is 3s. Or Ifrit doing the same in-between autos. All the while they keep the same clunky delay on abilities that we order them to do, that we've been trying to get fixed since ARR's launch. That's what Bahamut does currently. It would actually be a step backward, not forward, to implement how he works into their kits. The one thing that's near-universally agreed on is that the Egi overall don't feel impactful, but I don't think embracing Bahamut along with his flaws is the way to go about fixing that problem though. I've talked about it in my prior posts in this thread, however, so I'd advise reading those instead for more details.
For the Summons overall, simply consider two Quality of Life adjustments to how they should work. Make all Summons, including Bahamut, auto-cast the way Bahamut currently does, but only when you use an appropriate spell. Not on any oGCD skills whatsoever, unless they specifically state that they command your pet to do something, like Devotion and both Enkindles do. All of the auto-cast spells would be instant, and have a 1.5s recast timer. Ifrit/Titan would not auto attack either. Similarly, make Bahamut root himself the same way Garuda does. As long as he is in range of the same target you attacked, he shouldn't move, at all.
Another thing that would be nice is if the text for Damage over Time and pet damage from a pet that belongs to you showed up the same way GCD and auto attack text does. With the ability to delineate between those types by changing their colors if you wanted, provided they were your own. That way you could see the damage your DoTs are doing and whether they're critting or not. I guarantee we'd appreciate DoTs and Pets more if we could see their damage flying up constantly alongside our own. In general it would be nice to fiddle more with the floating damage/healing text of this game, to fix issues like that.