I presume this is the identity you had come up with that you said I skipped over, yes? Well, I will say that I personally do in fact like this concept of what a fighter/wizard's story for a class could be. Heck, the video game Dragon Age has an entire order called the Templars that have this function in their game world... keeping the apostate Mages in check. Now granted, I don't think Templars actually use magic in their duties themselves if I'm remembering correctly (and I could easily be wrong)... but the idea of the class story still holds a bit of weight.I would possibly propose building the swordmage, ironically enough, as an anti-mage or spellbreaker. Magic and mages in the world of D&D are dangerous, and the swordmage is equipped with both the swords and the sorcery to shut mages down, bypass spells and arcane wards, and resist magical monsters.
The only issue at this point then is whether the setting of D&D feel as though they are missing the identity of the "magic police" (for lack of a better word). If this was a trope that D&D needed, then an arcane half-caster that specializes in counterspells, dispelling magic and so forth could actually become a "thing".
If you were to ask me personally if I thought this identity of an arcane half-caster would be embraced... my answer would be "probably not". Because the story doesn't dive into the idea of what I think most people have of an fighter/wizard arcane half-caster just like the Bard and the Artificer don't. But at least this class HAS a story, so props to you for actually coming up with one.