We've given both. A bunch of examples. You've shot all of them down.
Mechanical differences from the base gish classes/subclasses:
- Martial Arcane Half-Caster (wizard-version of a Paladin) that gets Spellcasting and a Fighting Style at level 2, most/all armor and weapon proficiencies, and Extra Attack at level 5
- A "Spell-Strike" ability, where they somehow merge offensive spellcasting with weapon attacks (putting lightning into Thor's hammer, teleporting when you attack to lock down enemies as a Swordmage, etc)
- Using your main weapon as a Spellcasting Focus automatically
- Subclasses that promote different styles of being an Arcane Gish (Swordmage, Duskblade, Mage-Slayer, Witcher, Elementalist, etc)
Thematic Differences:
- You are especially trained to perfectly merge spell and blade. You're not like an Eldritch Knight, which is a fighter trained in some magic, or a Bladesinger, which is a Wizard trained to have some martial capabilities. Your training was unique and specialized to properly integrate both spellcasting and martial fighting into your style of fighting, making you an elite arcane warrior (typically existing in societies that are both militaristic and have some style of arcane training, like Netheril, many Elven societies, and similar cultures)
- You choose a specific discipline to combine your style of fighting with a style of spellcasting. You could be a Mage Hunter, a type of bounty hunter that is specifically trained to track down mages and use your magic to counter their abilities, or an Elite Bodyguard that uses both Abjuration magic and physical protection (Shields, Armor) to be an arcane bulwark that protects your liege from harm, and so on
- You are to the Wizard as the Paladin is to the Cleric and the Ranger is to the Druid. You're trained to master more Arcane types of spellcasting (evocation, abjuration, necromancy, anti-magic, conjuration), instead of the Ranger's focus on nature magic (entangling prey, goodberry, pass without trace) or the Paladin's focus on divine magic (healing, warding off the unholy, resurrection).
- You don't have to swear an oath or live in the wilds to get your powers, instead, you have to undergo rigorous training and possibly even bodily modifications to properly merge spell and blade. You didn't study a book like the Wizard, and didn't just learn martial training like the Fighter, but instead learned how to unlock magic through certain movements of your blade as somatic spell components, or arcane war chants that act as verbal spell components, or using the magic of your blade to fuel your casting (allowing it to be a spellcasting focus, ignoring many material components of your spells). It's a long, grueling process that takes years to master, and makes up for the normal restrictions of getting similar powers (becoming one with nature, worshipping a god/swearing an oath, unlocking the secrets of the universe by spending a decade in a library, etc).
Is that really not enough? You said that you liked the idea of an arcane gish (until we somehow convinced you otherwise), what kind of difference do you think would be good enough to set it apart from existing classes/subclasses?