Let me put this simply.
Training + Supernatural power source = Supernatural effects.
Training + NonSupernatural power source = mundane effects.
If Fighter has supernatural power source, then they get supernatural effects. If Fighter does not have supernatural power source, they get far less amazing effects.
Supernatural does not mean "magic". It DOES mean something more than training though. Psionics, supernatural bloodlines, otherworldly blessings, channeling nature, focus/ki/mystic life force. The barbarian's rage is supernatural and more than just a bad attitude. They channel Primal energy into extraordinary abilities. If Rage were stated to be even more overtly supernatural, it could get away with even more supernatural effects. Is rage magic? No. Does it cast spells? No. (Well, depending on the subclass). Is the supernatural? You bet your World Tree! And there is a lot of design space the barbarian left on the table because they were too afraid to say rage is overtly supernatural. The same is true of the monk's ki/focus abilities. Monks were overtly supernatural and WotC dialed that back some. (I get their reasons, but still). Monks can speak any language! They stop aging and are immune to poison and diseases! Why? Frickin mystical energy.
Monk, Barbarian, they are on the cusp of getting it. Dial them up a few degrees and then the mountain-smashing can begin! All you gotta do then is give fighters and rogues the same treatment and they can join the party.