Just change the spell list to "powers". A wizard/cleric/druid/sorcerer can cast any powers on their powers list with spell slots. A ranger / fighter get some of the powers as an at-will or x per short/long rest. Monks use ki points. Certain monsters (and perhaps certain future class options) can cast them psionically, etc.
For certain classes the power wouldn't be treated as magic. Wizards continue to benefit from being able to draw on a wide variety of supernatural powers by weaving magic with spells. Others achieve certain powers through force of will or because they are supernaturally gifted.
I like how in Flee! Mortals (by MCCM) they distinguish psionic vs magic power and use "supernatural" to apply to both. Psionic powers are not affected by dispel magic, counterspell, anti-magic fields, etc. Similarly, in my ideal system, a power used by fighter, monk, or ranger would be similarly unaffected by anti-magic. Only if a power specifically states immunity or the ability to counter psionics, or any supernatural power, will it affect non-spell powers.
I would go one further. I would like more granularity in anti-magic. I would like to see some anti-magic only affect arcane magic and not divine magic. You need clerics to counter clerics as a battle of faith. Wizards are better and countering wizards, because they know each other's tricks. I don't know how well that would go over with the wider gaming community but I'm thinking of making some homerules for this. It really wouldn't make the rule much more complex in terms of being able to remember them but it would add to complexity in terms of strategizing and tactics, which would make combat and certain challenges more interesting in my opinion.