We don't get Ramuh, leviathan et al because we are unable to absorb more than 3 base types of Aether. We have Bahamut because his aether was infused into everything and we have attuned to it. As Bahamut and Phoenix aether was dispersed at the same time during the calamity (Bahamut's on his death, Phoenix to restore Eorzea) we also phoenix's aether. Though it is not said, directly this is the conclusion I came to (in regards to phoenix)
Trances were a way to circumvent the inability to internalise more than 3 types of aether. The lore book 1 states
"Dreadwyrm Trance
Certain egi require an impossible quantity of life force to manifest as a separate entity. This technique was devised to circumvent such mortal limitations, allowing the summoner to harness the power of the elder primal Bahamut by infusing his own being with the essence of the mighty Dreadwrym."
While egi's are stated as
"EGI is the name given to conjured avatars of the summoner. Only a practitioner whose aether has been stained by the essence of a primal can thenceforth shape their own energies into a willful embodiment of the original entity's dominant aspect."
Terraflare (LB3) is explained as
"Terraflare: an offensive magick that echoes the ruinous blasts of light unleashed by the Elder Primal Bahamut. All those who Bathed in the dreadwrm's scattered aether during the Seventh Umbral Calamity possess the latent potential to master this mighty summoner spell."
All of these go to explain why we gain the skills in the order that we do. The reason, I think, we cannot summon Demi-Alexander is that we never actually defeated it fully. There was no implosion of aether and therefore nothing we could trance to achieve the required aether to create a demi primal. Alexander lost to us, but he was a projection. It then sealed itself off outside of time thus meaning there is no aether to draw upon. At least that is my take on it.
All in all, the lore around the job makes sense. The biggest conflict seems to be in the ideology of "what is a Summoner?" Traditionally it would summon sentient beings to do its bidding. In XIV we summon a shade with the essence of the being, it can look the same and possess similar ability but ultimately it is a simulation and as it is a creation using the aether of the summoner, it can never truly be as powerful as the original. Demi Bahamut and Phoenix, work slightly different as it is the channelling of the aether around them to achieve the trance and then the summon. (Not sure how this works on the first though - Osmosis I guess?)
Overall I am not saying your vision for summoner wouldn't work, but I think it would break the lore of the job. Summoner currently has its problems and the root is in too many systems that on their own are fine generally, but when brought together they make an incredibly clunky and busy job. My own thought on resolving this isn't a complete overhaul, but rather the streamline or removal of one or two of these systems and the optimisation of the others to make it flow better. Where I to choose, it would be dots that went or were adjusted and perhaps aetherflow/energy drain - though I do like this system. My reasons for this choice is that we have moved so far from arcanists. Dots don't tie into our kit much, they are just there. I think the change with ruin was an attempt to bring cohesion but it ultimately failed. Not because the idea was bad but because it was another system on top of a large load of systems with too many 30 0r 60 sec cooldowns. Taking them away would ease some of the clunkiness, but also alleviate the busy feel of the job while retaining a lot of its core functionality. There are of course other ways to achieve this. But I think this would be the simplest way. Also if hey rolled the damage from dots into Bahamut/ phoenix / deathflare I think it would make those segments feel much more rewarding, at least in theory.