When I first heard about Eberron I was enthusiastic. Steampunk D&D? Yes please. But then it turned out to be high fantasy so I made it work in Ravenloft instead.
So I think I don't like Eberron. But then, I also didn't like FR in 3e either. It was much too high fantasy for me.
In 5e I like FR just fine. I think it is partly due to it being the core setting so they need to pull things back a bit. But moreso I think it is the edition itself. When orcs are still a threat to 10th level characters the world is different. When you don't have merchants selling goods for coppers alongside others selling magic items for thousands of gold pieces, the world is different. That comes through.
To start with, you're using 'high fantasy' in a slightly different way than usual sense. Tolkien's Middle Earth is high fantasy, but it doesn't have magic item market places and even qausi-angels walking the earth still have to run away from large enough hordes of orcs.
I'm guessing you mean something more like high-magic or high-power. And, yes, Ebberon and FR are both very high-magic settings. 5e may mute that a bit with it's core assumptions with regards to magic items, specifically, and certainly dials down the range of competency (and thus power in some senses) via bounded accuracy.
Using FR as the default setting forces it to be a little less crazy, too, I'd agree. But only a little since 5e is very high-magic as presented (all classes, and 33 out of 38 sub-classes, in the PH have some sort of magical powers), just not automatically high-magic-
item to the point of having assumed wealth/level or magic-item economies.
I am no Eberron expert, so I ask, could a 5e version keep the feel and flavour of it while also being grounded in traditional fantasy tropes?
Eberron is very much a steam-punk setting, with magic taking the place of/providing rationales for some of the de-facto technology instead of literal steam. You have lightning trains instead of steam trains but they're still trains, that kinda thing. You can't really have 'low magic' Eberron without losing that steampunk vibe, but you could tilt the campaign more towards that kind of quasi-technological magic rather than traditional D&D spellcasting and " ____ of _____ " magic items. It'd be no small amount of work, since you'd be creating a lot of gear tossing out most of the 5e classes, but you could do it.
I am curious because the big difference here is that Eberron was made in 3e so maybe its world was modeled on the conceits of the edition whereas FR has reverted to its older roots?
A central conceit of Eberron was that everything in D&D had a place in Eberron. That's not exactly contrary to 5e's philosophy with regard to other editions, and it's not like 3e threw out a whole lot from prior eds. FR has gone through some fairly lame convulsions in its history and some retroactive continuity, but it's still the same very-high-magic D&D setting it was in both 3e & AD&D.