Since the dawn of time (also known as my early days in gaming), magic-users couldn’t have swords.
I’m talking OD&D, B/X, BECMI, AD&D 1E, AD&D 2E, 3E, 4E, 5E. In the later editions, you can take a proficiency to use one but it’s blocked out of the gate.
They are proficient in dagger use though (which is a teeny sword you have to get even closer to use).
Anyway, I digress. I’ve never seen the need to block a wizard from having a sword from the get go (or any melee weapon at all) as there is inherent risk in a wizard going anywhere near melee that you may as well let them use what the hell they like. SQUISHY UNARMOURED MEATSACKS.
Gandalf and Glamdring (bastard sword methinks) is the stereotypical example of why I want wizards to have swords.
I say let them, Magic-users (and later Wizards) are so squishy, letting them use a sword is delegation of risk, and a last resort as a player.
My main point here is: If you want a soft, squishy character to have access to melee weapons that the rules don’t allow, let them. It’ll be more fun when the giant rips off their heads and squashed out the innards like a tube of toothpaste.