Paul Farquhar
Legend
No matter what I write for my backstory eldritch knight doesn't feel right because he is just as much of a mundane fighter as everyone else IMO. The cantrip change would do that, but there are other subclasses that have a similar effect?
Well, I chose a race that had a cantrip at level 1 anyway (Genasi). EK is kind of archetypically a High Elf. Obviously Arcane Trickster is similar.
Variant humans could get Magic Initiate or Ritual Caster at first level if it made them feel the character worked better that way.
I don't see an issue with barbarians, your barbarian probably has shamanistic beliefs from the start, and they only gradually become more magical, and that depends how you fluff it. A bear totem barbarian doesn't change into a bear, they are just as tough as a bear.The same issues with eldritch knight tend to apply for many barbarian subclasses as well.