ADC means "Attack Damage Carry". A good team comp should have a balanced mix between attack damage, magic damage, and tank.
If a Mage were to go ADC, that would mean instead of having an ADC, you would have more of an APC (Ability power carry). This usually means that your team comp would be very AP heavy, dealing little AD.
The problem with this is that one or two MR items is all the enemy team needs to completely shut down the majority of your team.
If you had a proper ADC this wouldn't be the case, since the ADC would shred them if they don't buy any armour, so they are now forced to buy both MR and armour to survive, which if they do buy both, it would take up a lot of their build, forcing less room to buy damage items.
Now, if you were to look at this realistically, in most Bronze/Silver matches a proper team comp is the least of your worries. Chances are, the enemy carries will refuse to buy anything that doesn't give them damage, so you will blow them up even if you are all AP.
If your question is why you don't see many people playing mages in the bottom lane, I just want you to know there is no rule saying what you can or can not play in each lane. People just aren't used to the idea of AP champions being played as ADC so they are afraid to try something new.
One gameplay related reason that does come to mind, is that the damage Mages deal can't always keep up with the damage typical ADCs deal. Most ADCs can just right click and deal semi-consistent damage early game, and heavy damage late game.
Mages require mana to deal damage. Early game, you won't be able to do much as you will run out of mana fast if you use your abilities a lot. This makes it very hard to win lane, as most of the time your enemy laner will have more CS than you.
Late game, Mages will have sustainable mana, so they can usually spam abilities in fights, clear waves, ect. and not have a problem with mana. But they will still have a hard time keeping up with the damage the enemy ADC is doing.
I could also talk about counter-picks, but I didn't wake up planning to write a 1000 word essay.
Tl:Dr: Mages being played as ADC makes it easier for the enemy players to build defensive. In Low-ELO this isn't a problem most of the time, but smart players will be hard to play against. Since Mages rely on mana, they will usually have a hard time keeping up with the damage that ADCs deal. (Unless they get fed of course)
EDIT: Forgot about killing turrets, but that's also a factor. Sorry for the wall of text btw...
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