Not necessarily. I mean, Wizards can cast Spirit Shroud, and while they could get a lot of damage out of it with the right spell, having to be within 10' of the target is not something most Wizards are built to do. In fact, I'd point out that when the "Bladesinger"* cantrips were introduced to the game, despite the fact that a Wizard can certainly use them, the problem was mostly other classes getting their hands on things like Booming Blade for a damage boost than Wizards suddenly waving swords around!
Wizards can cast Enlarge to increase their melee damage. They typically don't. Wizards could cast Haste on themselves for another attack action. If they were to use the spell targeting themselves, that's probably not what they'd be doing with it. The problem isn't the Wizard having access to spells that a Gish would want. If anything, that's to the benefit of Eldritch Knights and Mystic Theurges, spell poaching Bards or multiclassed Wizards. The problem is that someone decided that "melee Wizard" wasn't some weird fringe build that would take a lot of effort to make work, and instead created a "melee Wizard" subclass complete with Int to AC and concentration saves, normalizing the idea.
If Bladesinging is allowed to exist, you're never going to get a good Gish. It's that simple, because the Bladesinger is allowed to masquerade as a melee combatant, even getting Extra Attack (something even Clerics and Rogues can't have, for some reason), and should that somehow prove not viable, they're still able to cast all the best non-melee spells in the game, up to 9th level, with boosted AC and concentration saves on par with a Sorcerer. Or better, should they decide to take Resilient!
Even if you made a 3/4 caster who can mix it up in melee, the fact that they can't suddenly decide to True Polymorph into a Marilith and get access to spell levels more slowly will hamper their power. Plus, the Bladelock is right there, already living in that space! And a 1/2 caster, the so-called "Arcane Paladin" would even be worse off, I'm sure, as they'd have to** should be a weaker warrior than an Eldritch Knight!
*I find it ironic that they are all based on Swordmage powers from 4e.
**Well, if anyone cared about balance at WotC. The Bladesinger existing makes the EK seem like a bad joke. Especially in 2024 where they're not even allowed to use Action Surge with the Magic action!