I think the way they should do it is have the X-gene exist in the MCU but laying dormant. Mutants exist but are very rare so you can have Xavier, Magneto, Wolverine (Chris Evans WW2 flashback cameo) and Apocalypse who are already established but hidden.
Have the second snap from the hulk or multiversal tampering activate the X-gene and explode the mutant population in the MCU. It forces Xavier and Magneto into trying help and guide the mutant population while also letting you easily integrate versions of these characters so the audience can adapt to this change within the MCU with the characters.
This is it. The snap and the multiverse can play a role in activating the X-gene, but the characters have to be from the MCU's 616 earth.
WandaVision already said that Wanda had powers before Strucker's team experimented on her. The Mind Stone basically unlocked her powers further.
It seems the bracelet did the same for Kamala Khan.
Mutants have always been around, most of them have their powers dormant as you say, but interaction with the infinity stones or other artifacts are what's unlocking their powers right now in the MCU. Others might've had more access to their powers, but not quite fully.
The Snap extended that interaction to the entire universe, thus more mutants will have their full powers unlocked, and younger/newborn mutants will now work as they do in the Marvel lore.
In Magneto's case,
PlanetSmasher 's suggesting is pretty good. He could've been one of the mutants that had their abilities manifesting in a more powerful manner, so the Nazis took more interest in him. Hydra could've exposed him to the Tesseract or something.
I lean more into the freezing him thing. Like what if Hydra tried to use him like the Winter Soldier and he fucked them up real bad? You could have him escape sometime in the 90's... Live with very limited powers, frustrated and embittered by injustice in the world, literal fascists returning to power... and his powers not being enough to do much about it.
Then the snap happens and he returns with full Magneto powers.