Hm. It's been more than a decade since I've played MtG, so consider this more of a thought exercise.
But what if you just accept that it's going to run like a D&D campaign set in Dominaria? Really, I think that's a good place to start anyway, since if you get too weird, you run the risk of alienating players. Instead, if you want to make it like MtG, why not focus on what makes Magic so... magic-y?
In other words, why not tie the characters to the cards?
Okay, so your background is tied to a land card. Are you from an island? Throw an island card onto your character sheet. Ditto for mountains, forests, and the like. And I'm sure there are still a lot of weirder land cards that PCs could hail from. Whether you want those cards to be represented by traits is up to you.
You can throw in spells as cards, and give them to players. Rather than putting fireball on your character sheet, you have a fireball card. And of course, when you cast it in play, you tap it.
Magic items become Artifacts, and summons are now actual cards that the summoner has access to.
If you want to be really creative, you could have players actually have a "deck", and they get to play one card each turn in addition to their actions - laying a land card, channeling energy, and performing various spell effects. You could devise some sort of subsystem using magic cards to correspond to RPG events. How you'd do it would take a lot of work, but it's doable, I think.
What I'm getting at is, it's a losing battle trying to make the setting of M:tG work if you divorce it from the cards. It'd be like trying to convert Greyhawk into a diceless setting. Or converting a record like Dark Side of the Moon into a novel.