Several come to mind as being magic-rich...
RuneQuest (any Chaosium edition, including the jointly done GW and Avalon Hill 3rd) has most characters being able to use a spell or two. The system is percentile skills, 3-18 attributes, low hitpoints, no levels, and optional hit-points by location. It is, however, premedieval - Bronze age. So it's a rule-out. It is, however, the same base engine as Elric!
Probability of Match: appears Previously ruled out.
Recommend: high. RQ3 can easily be run post-middle ages by adding guns. But see Legend.
Which leads to some other such games.
Legend would be a good choice for what you're looking at; it's Mongoose's flavor of RuneQuest minus the Gloranthan setting. Very workable, but Guaranteed not what you played, as it's a 2011/2012 game.
Probability of match: Nil
Recommend: Moderate to strong. If you liked Elric, it is close enough to be familiar.
Also related to RQ is Hero Wars also know as Hero Quest. Being Gloranthan, it's low-powered but all-pervasive magic. It's also bronze age, but that can easily be missed or ignored. System is purely d20's.
Probability of match: Very Low
Recommend: Moderate - very narrativist.
The Arcanum (The Atlantean System, by Bard Games) has a lot of magic using types, and many combattant types have magic as well; the setting is magic-as-technology in a fantasy Atlantis, Lemuria, and Mu. Been out of print for quite a while. 5-20 stat range, 1d20 combat rolls, percentile skills, class & level advancement with ability to buy extra skills.
Probability of Match: Low.
Recommend: Low. I like it, but most don't.
Amazing Engine by TSR, often called Amazingly Bad Engine, was a passable game, but had some excellent worldbuilding for the setting books. Long out of print, but due to the mediocre mechanics, inexpensively if rarely obtained.
probability of match: moderate
Recommend: low - system is playable but not well liked.
Rolemaster, HARP: ICE's various Rolemaster flavors; HARP is essentialy RM Light. All do what you're asking, and HARP seems to do it more simply, but the stock settings lack the explicit disdain of mere fighters; the cultures of the lines' fans do, however, tend to disdain pure arms-realm types.
Probability of match: Nil. Previously Ruled Out
Recommend: Moderate to high - it does exactly what you're asking for, and does it quite playably, especially in this era of spreadsheets.
The Fantasy Trip is an older contender. Long out of print, you'd know the books as Melee, Wizard, Advanced Melee, Advanced Wizard, and In The Labyrinth. 6-20 stat range at start, stat driven, Xd6 for stat or less. Lacks the disdain, but does have magic to the technology point. Written by Steve Jackson, who later wrote GURPS, it's somewhat similar.
Probability of match: Low-to-moderate. It can be played very high or very low magic.
Recommend: Weak recommend. Hard to find, not quite right.
Tunnels & Trolls is the final contender to offer. System is 3-18 stats at start. Lots of casual players have disdain for fighters, as wizards have more obvious inherent XP gains, and it's a unified XP table. 4 classes (Warrior, Wizard, Rogue-Wizard, Warrior-Wizard), no skills until recent editions (5.5, 7.0, 7.5). Published adventures, both solo and GM'd, are aimed at high magic parties and frequently go way ver the top.
Probability of Match: Moderate.
Recommend: High. I love this game. Originally published in 1975... and been in print most of the time since then.
Monsters! Monsters! Variant T&T, playing the monsters rather than the men, elves, dwarves, hobbits, leprechauns, and fairies.
The "Not Able To Recommend" Maybe possibles... Mostly because I don't know enough, but by listing them, if correct, they'll spark the memories.
Arduin is a magic-heavy D&D variant that became a unique non-D&D system in later life. It's also the name of the setting attached to it.
Skyrealms of Jorune is a system that I recall, read, and sold. It was a mixed magic and tech setting. I can't speak much more.
World of Synnabar is similarly high magic, and didn't catch my interest.
Rhand: Morningstar Missions by Leading Edge Games. I sold my copy off, but ISTR it being percential skills, 3-18 stat range, post-holocaust on a colony world, mixed magic and tech.