level 1 ganks would suck. if the enemy casts a q, a w, and an e on him, assuming he doesn't die, he could potentially use all 3 of those abilities.
Also Cooldowns would be a problem. because to be balanced each ability key needs a cooldown that matches the ability used, such as Pantheon's spear versus an ability like Blitzcrank's grab. if a blitz grabs him, and he uses the grab too, then a pantheon q's him, will he have the spear immediately, or will his q still be on cooldown.
Another problem to nail down is his mana pool. the amount of mana a champion gets is proportional to the abilities they use, and therefore, a champion who uses many different champions abilities would have a very, very hard time with mana issues.
Also, he would be completely offensive, having limited escape options, because abilities that only affect the champion that casts them, or their allies, could not be used be him. so he would never have any heals, or any attack speed buffs, or movement speed buffs, or any sort of buffs.
Also some abilities are based on whether the champion is a melee or ranged. i.e. Jax's w, or his ult for that matter. imagine how OP a ranged jax would be. or any ability that jayce uses would make his abilities really interesting because it would look absolutely ****** to see a ranged squishy champion leap on to an enemy or a melee tanky-ish champion kiting enemies with abilities.
Scaling is another problem. if an ability he uses scales with AP and he has a bunch of AD, he would do little to no damage with it. or vice versa, unless you made it do the same damage that the enemy did, and then he wouldn't do as much damage if he got ahead of the enemy, making it almost impossible to win a lane.
Anyway, i don't think a champion like this would be practical at all. If anything, he would be a very, very, situational pick and would be retardly hard to play well. I like the idea a lot, i just don't see this ever happening, unless you only made it the ult that was the mirrored ability. that could be interesting.