If you add AP stats to fighters, it doesn't end well. Its either too weak to be useful, or too strong to be balanced. Which led to things like AP Trynd. It was not a good thing. Your E had, and I believe still does, a 1:1 AP ratio, but the AP ratio on your Q was much higher. Which meant, you did loads of damage, healed massive amounts, and then had your ult, which gives you immunity to death. So essentially, after scaling up enough, you couldn't die.
The reason why late game mages would dominate, is that items would become more important, since they have more power vested in them. Since late game mages don't rely on base damages, they rely on the items they pickup. And, since those items are stronger, they are stronger. While they would be weaker early, so would everyone else, the difference is they would scale better. It would be the, "Stall till 30 minutes, then gg" kind of situation.
Junglers are seperated into several categories, but the ones I'm talking about are early game, and late game. For example, Elise. Elise has high base damages, but falls of over time. If you slash her base damages, while buffing item, which she doesn't scale as well off of, you are nerfing her. Conversely, someone like Yi. Yi doesn't rely on base damages, instead uses items to increase his power. If you slash his stats, he's weaker, but since he scales better off of items, he is overall stronger.
With gold leads, since base damages aren't as strong having more gold, and more items, than your opponent, sets you at a much higher advantage. Therefore, gold leads are more important, and more rewarding.