The current non-magic archetypes are:
*Brawny Guy.
*Agile Guy.
There's a few archetypes that might make you want a little Int, Wis, or Cha, mostly for skills, but the occasional thing like Battlemasters granting temp hit points or Arcane Tricksters wanting decent save DC's do exist.
It's all fairly vestigial however. You can certainly create a non-magic guy who isn't brawny (Dex based Fighters, Rogues), or a non-magic guy who isn't agile (Fighters with heavy armor). About the only ability score you'd probably not want to do without is Constitution...maybe.
In a group with ample magical healing, maybe a Fighter with only average Constitution would be fine.
You can even make spellcasters without a heavy investment in their spellcasting ability, depending on your spell loadout.
Sometimes a class will reward you greatly for having a certain ability score (like Charisma for Paladins), but not always. It's always struck me as an odd way of doing things.
I remember playing in AD&D, when every ability score granted some kind of useful advantage (at least in theory), no matter what class you were, but even then, some were weighted more than others- only Clerics and Druids got bonus spell slots for high Wisdom, for example.
I remember playing Earthdawn, and I was really impressed how every attribute added something to your character, from basic stuff like carry capacity for Strength, to movement speed and Physical Defense from Dexterity, Magical Defense for Perception, Magical Armor for Willpower, Social Defense for Charisma, etc. etc..
Or when I played games like Stormbringer, which had secondary characteristics that were derived from several different primary ones.
I don't see D&D going in this direction, but it would be nice.