IIRC one Gnome and one Elf subrace have a single cantrip, Drow and Tieflings have more spells. So you still have Human, Dwarf, Wood Elf, Halfling, Rock Gnome, Half-Orc, Half-Elf and Dragonborn without magic. Or you can play Forest Gnome and High Elf as well and ignore just the bonus cantrip. There are also many non-core races without magic.a no magic 5e game is basically use the combat rules and forget all races but human and all classes but fighter exist.
Fighter, Barbarian, Rogue have enough archetypes without magic, especially if supplementary books are allowed.
I wouldn't even remove the Totem Barbarian, for what its 3 spells are worth, I would narrate them as non-magical abilities and keep the archetype available.
Also I don't think Ki is necessarily magical enough to disqualify the Monk. The truly magical effects are from Empty Body, until you reach level 18 there is not much to worry about.
So you have easily enough races and classes for a party of 3-4 PCs without overlapping.
Not that the game would be unplayable even if everyone was a human champion fighter...