Why are all AP champions burst?

Seriously, if you're an AP scaling champion, you're either an assassin or a burst mage.

Even the tanks tend to be burst focused.

The riot releases a champion line who is intended to be a kind of, DPS mage, yet his strengths are still when he's bursting an enemy. Why is this?

I think it's pretty easily broken down. Even if you ignore the idea of something like "Spell crit" which is what gives most auto attacking champions their burst damage. AP scaling lacks one fundamental feature of AD which lets some AD champions be played reliably as consistent damage champions. Lifesteal.

Yes spell vamp exists, but it's been in a pitiful state for years. It's been nerfed to the point of obsolesces, all because a few champions were far too good at utilizing it. The reason i point towards spell vamp being the key factor here is because of how kiting champions tend to win. Say an ADC like ashe, she is powerful via her ability to attack move. Having lifesteal on her is extremely powerful when fighting melee champions, because it allows you to heal back the damage the enemy did for the period they were in range to attack you. This means that they can't simply chase you down while jumping onto you a few times to get chip damage in. Like people can do against sol.

I'm not going to say that sol is bad, hell i've already gotten a penta kill on him, but he fails as a consistent damage mage. He has no form of sustain for when a champion has damaged him, and having a dead zone (standing on top of him) in the majority of his kit, without having some form of healing for when he isn't in range of the enemy he's made to be more effective as a burst mage, because he's simply more reliable as such.