Is there some game balance concern at work here I'm not seeing?
Intention of the rules
Mages cannot wear armor, have only few hitpoints and are very restricted concerning weapons. This is for game balance reasons to offset their enormous magical potential. This balance considerations are explained by the fact that mages have no training at all with weapons or armor to focus on magic. They can only use very basic household weapons and a crossbow is not one of them.
In reality, although we consider crossbows a relatively easy weapon to use compared to the bow, that's only half the truth. The mage's weapons, like darts, staff and dagger can be handled be virtually anyone. I can handle them good enough to not hurt myself (mostly...) and I have no training at all. A crossbow does need training. Although I theoretically know how a crossbow works, if you gave one to me, I'd probably have problems to get the mechanic to work. Is it easy to learn? Probably. Does it need training? Yes, it does. And why would I train to use a crossbow when I could spend the time to learn magic.
Crossbows are a relatively powerful weapon. However, compared to the bow using 2e rules, it's horrible. There is nothing in the books actually describing the crossbow as inferior to the bow, so I guess it was not intended for it to be that weak. It did have superior range, it's just that D&D does not actually use that range very often. If the next dungeon wall is 30 feet away, having a good long range is not enough to make up for the advantages of the bow. The crossbow did get stronger in later editions. So the reasoning for not allowing mages to use one probably was that the crossbow should have been a powerful weapon at least somewhat comparable to the bow and therefore the balance would have required it to not be usable by mages.
I think from a pure game mechanical point, crossbows being as weak as they are, having one as a mage would probably not be too powerful. However, the crossbow does not count strength penalties either, so trying to use one with a strength of 6 would be pretty powerful compared to all other options. Gameplay wise, it doesn't make a difference with all the other combat disadvantages of the mage (even if it grants more damage, I'd say let him have it for the one occasion every session the mage actually manages to hit), but personally, I would not allow it for style reasons. Have you ever read a book or seen a movie where the mage was dragging along a crossbow? As a matter of fact, I cannot even remember one where the mage had any ranged weapon at all.