So, we're back to a game design discussion, again.
The short version is, if it doesn't make sense to you, don't use it.
Squishy wizards are almost more of a gameplay consideration. If you have a game, and you're balancing ranged damage against melee damage, if your ranged damage units do enough damage, you can create a situation where melee damage straight up doesn't work. It's not viable. The 40k meme about the Tau comes to mind: “Sure, they suck in melee; too bad you'll never get there.”
If you tone down ranged unit's damage, that can easily create a situation where they become the ones who are irrelevant. Such was the experience of every level 1 Wizard in AD&D. Once in awhile, you can get into the perfect situation to end an encounter, but most of the time you're just biding your time until you get to level 5 and can learn to accidentally fireball your party's front line, but that is a long time from now.
If ranged units can do a lot of damage, they need to be fragile enough that you can remove them from the board. And the Tau comparison comes back to mind once again.
All of this combines to create a board environment, where melee fighters need to be tanky enough to get into combat and stay there. Ranged units need to be fragile enough that they can remove each other, deal enough damage to harass the melee units, without doing so much damage as to render them completely irrelevant to the board.
And, while you can build a story around that structure, you don't need to.
Gandalf isn't a fragile wizard. He's not some “book nerd,” who spent high school getting shoved into lockers. When the time comes, he goes toe to toe with a Balrog (or, the Balrog, whichever), and doesn't immediately die. He clearly manages to hold his own, in melee combat, with a massive monster. (In fairness, he's also not human. I mean, none of Tolkien's, “the race of men,” are conventionally human, but Middle Earth's Wizards are an entirely different race of beings.)
In a lot of games, solution is to give the frontline fighters a ridiculous amount of health. Now, I'm going to trash on D&D for a second, but consider that a 10th level Fighter should have somewhere around 94 - 114hp. Remember that critical hits represent some kind of significant injury. These are not just blows that connect with your armor and will leave a bruise, this is someone ran you through. Someone could crit on your fighter, with a long sword, and stab them in vital places at least 4, and probably 5 times, before it actually kills them. That's a comical amount of damage someone to suffer. (Now, granted, a 10th level character in D&D is basically a superhero. If you're thinking of Boromir's death in Jackson's Fellowship of the Rings, that is what it takes to put down a relatively high level fighter in D&D. Which is to say, hilarious amounts of abuse.)
If you signed up for that, cool. I'm not going to stop you. I'm not even going to tell you it's wrong. If you want to tear down a super-humanly powerful character through prolonged combat sequences, or due to attrition of multiple fights in quick succession, that works. I mean, hell, that's how DC killed Batman in the 90s.
If your wizard power fantasy is that a wispy intellectual gains cosmic power through hard academic study, cool. Again, that's entirely valid, and as I mentioned, it even fits into a power fantasy. If you were bullied as a teenager for your atypical interests, and habit of reading, here's a character that studies strange and esoteric subjects, and has real power as a result.
At the same time, it's entirely reasonable to have an averagely healthy mage, whether they study magic academically, or have some ingrained talent that they've honed, plop them down next to a veteran swordmaster who's fought in wars on nine continents with the scars to prove it, and while they may look a bit anemic in comparison to their buddy, is still in better shape than the average villager they interact with on a daily basis.
That's where I tend to land in all of this.
When you're creating characters for your writing, it can be helpful to assign them attributes. Now, I don't mean this in the literal RPG stat blocks. (I've tried that a few times, it doesn't really work for me.) But, just a few text descriptors (which, does sound like Fudge, come to think of it.) You might describe your mage as Smart, or Intellectual, Wise (or Absent Minded), Willful. You know, “wizard stuff.” If you describe your warrior as, Strong, Tough, Tenacious, and Cunning, you're not making the wizard squishy, you're making another character less squishy. A lot of the time, we set the base line by what other people are doing. It's reasonable to say your mage is less durable than your soldier. (Unless your mage has a reason to be that tough. Maybe they're from some frozen wasteland, and are just absolutely jacked from surviving in a hostile environment.) But, that comparison doesn't mean that your mage is deficient.
Now, on the other hand, frail characters can be interesting. You're taking out their ability to fight conventionally, so when they do start decisively ending situations, whether that's through their own creativity and guile, or sheer magical power, it can be very gratifying. And, to be clear, I am very fond of flawed characters, especially when they have to work within the framework of their flaws to find solutions, rather than just overcoming them through the power of love, friendship and mescaline.
When handled well, flaws are about creating limitations for how your characters can solve problems. These can also make your story more interesting. If you say, my character can't fight, (and you don't back down from that and just let them cheat so they can fight, because they're so goddamn special), they're going to need to find other solutions. That can result in a better, more interesting, and less predictable story.