Aether as a fifth element was really more of an understanding of the circles of the world – Earth/Water/Air/Fire/the Pure Air beyond the Fire. Aether literally means pure air in Homeric Greek, though it took on a sort of mix of the Fire and Air properties later in connection to the Phlogistron theory.
The Ethereal Plane is not really that closely connected with this concept other than that Ether is sometimes considered the Fifth Element and the Ethereal Plane is the route from the Material to the Inner Planes. D&D's Ethereal Plane was really not so much a place as a mechanical explanation for incoporeal beings such as ghosts, and as a route to the Inner Planes to parallel the Astral Plane as a route to the Outer Planes. The Astral Plane is just as much connected to the ideas of the Quintessence as the Ethereal Plane is, after all.
4E got rid of the Ethereal Plane because it was trying to get away from un-places that exist as just-so stories and focus on planes where adventures can happen. This allowed the Plane of Shadow and Plane of Faerie and the Elemental Chaos to each ape a bit of the Ethereal Plane's shtick and divvy it up amongst them while allowing these planes to flesh out in ways beyond the vaguest of vague sketches past editions did for them. Shadowfell also got to ape Ravenloft for more setting details, and the Raven Queen/Shadar-Kai/Gloomrought/Dark Ones storytelling that was such an iconic 4E element really fleshed out the rest of it. I'd argue that Shadowfell was also influenced by Shadowmoor, while the Feywild was influenced by Lorwyn and the Elemental Chaos by Alara (all three blocks were releasing in the middle of 4E's design & development and release cycle. Though Innistrad and Eldraine would each emerge later to form more "pure" takes on the Shadowfell and Feywild respectively, as their own unique non-echo planes in MtG lore.
In -MY- D&D games, I consider the Ethereal to be of an elemental nature – by analogy to my Primordial Chaos being like a Quantum Realm Scale and my Astral Sea being like Intergalactic Scale; the Ethereal Plane is a sort of magical blue mists of wonky time and space that bridge the gap between the chaotic quantum madness of elemental waveforms and the more solid reality of the Material where enough waves are clumping up and interfering and entangling to become real matter with a real spacetime. And then at the huge scales of the Astral Sea, it starts to go cray again but in the opposite ways because we're dealing with immortals of unfathamoble sizes and lifespans, whose bodies are literally the plane(t)s around which the Wildspace systems spin.
So for me, the Ethereal really needs to be Elemental.
But I'd note that in D&D 5E assumptions, the Ethereal is ALSO connected to the Astral via those Deep Ethereal storms that cadn send you shooting up into the Astral Sea. So if you're doing a variant of the World Axis cosmology like I prefer, you may need to either jettison this idea or consider it as a sort of backdoor around the Material or as an exception that proves the rule.
I'd argue the reason it's not considered Elemental by the rules as a default is because of Sacred Cows attaching it to Incorporeal Undead.