Oofta
Legend
Sure they should.
Some monsters in 5e are immune to non magical attacks which means they cannot, which is the way it is for those monsters but I think that is a mistake for a game that does not assume magic items and generally assumes fighting monsters.
Specters are 5e ghosts that anybody can punch to death from first level on.
5e is a bit inconsistent on its monster design here.
It can easily be done.
In 5e turn immune to resist and non-supernatural martial classes can kill monsters with just a steel weapon. It will take twice as many hp damage to do so, but it can be done the same way they can defeat monsters with resistance to weapons.
To make it even more leveling across classes take away even resistance to weapons so Conan can kill the Spider God.
4e handled no-magic martial classes fighting monsters just fine, inherent bonuses put them on a fairly even playing field with stock magic item 4e treadmill ones.
I do a fairly low level of magic items, but if I were to do no magic items at all? I'd probably have silvered or cold iron weapons act as magical. As long as fighters don't have to have a caddy for all their weapons like we had in 3E.