For the most part I can agree conceptually what you are stating here and for you and probably others it works. But I'll say this for me personally... because all classes in 4E were on the AEDU format and all powers encompassed the exact same suite of grid tactics (adjacent damage, blast damage, burst damage, pushing, pulling, marking, knocking prone etc. etc.)... any so-called "non-magical" classes in the Martial power source did almost the exact same kinds of grid tactic actions that the so-called "magical" classes did.
Which means that at least for me I never felt any real difference between magical and non-magical powers other than the Martial power source not having ranged burst (AoE) attacks. Everything was just 'Move here, attack this/these creatures here, move those creatures over there, shift here' and so forth. None of it felt really different, even though we were supposed to identify certain ones as "maneuvers" and others as "prayers" or "spells". "Low-magic" campaigns and "High magic" campaigns did not exist as separate things, because for me the tactical board game felt the same both ways at the same time.
I do not doubt that other people did indeed conceive all of these powers differently and actually felt like maneuvers were different from prayers were different from spells. But unfortunately some of the rest of us did not. And that's possibly one of the reason why 4E maybe ended up being more divisive in certain circles than I bet anyone thought it originally was going to be and perhaps trying to recapture some of that 4E essence in 5E wouldn't actually solve the problems that people might think it would? (And not suggesting that's what I think you are saying, because I know you're not making that case.)
I'm not saying I don't see some simularities and that there was some differentiation coming from fluff (as in any edition actually), but there were a lot more mechanical "permissions" than just ranged burst AOE. Arcane, Divine, and Primal still had broader permissions than Martial. Martial powers rarely had elemental or other damage types, no teleport, no summons, no shapechange, no creation of objects like walls, no ability to petrify, etc.
What 4e didn't do was gate most status effects to magic only and made the "combat magic" play within the same resolution rubrik as martial -- attack rolls, HP attrition, most status effects available to everyone, etc. So there was a certain "sameness" there (although even in this level playing field magic did more).
And of course, a lot of the "high magic" was moved to rituals, an unfortunately neglected subsystem. I suspect a low magic game in 4e would remove rituals, wheras a high magic game would make them easier to find and use. By the end of 4e almost all the classic spells were in the game as rituals. If 4e had started with all these rituals in the game, gave a few free rituals per day to the classic full casters as they leveled up, and put their "rituals per day" as a classic spell chart then perhaps things would have been different...
I think the sameness feel for some comes from the fact that combat magic played on the same resolution rubric. Yes, you can teleport but you can no longer teleport everyone out of the encounter. Yes, you can shapechange but it gives you set stats that are within limits -- you can no longer scan all the monster manuals for something that is well outside any limits. You have to play inside the encounter box for the most part.
We've seen the version where everyone plays on the same enounter based resolution system, and I'd definitely be curious to see the version where every class (even martials) plays on the traditional magic resolution system -- where some of your abililities operate on the traditional resolution mechanics but some just bypass them (just work, bypass action economy, bypass skill rolls, etc.). And no, this would not have to involve giving spells to martials.