Damaging spells heal for x (+y / level) (+z%AP) health. Double this amount for non-damage-over-time single target spells.
By shifting the system towards spell application independent of damage dealt, you decrease the gap between single targeted and AoE spells. However, you open yourself up to a significant loop hole: Damage spell count and cooldowns now dictate sustain.
A mage such as Orianna has significant cooldowns, even more so burst mages like Brand or Le'Blanc. Similiarly, Anivia's Flash frost has a CD of 12 to 8 seconds and Frostbite has a CD of 5 seconds. Consequently, spellvamp is completely ineffective on burst mages as you intended. However, I doubt that it's viable on mages like Orianna. After all, there exists far shorter cooldowns in this game: Karthus & Cassiopeia
Cassiopeia: Noxious Blast 3s, Miasma 9s, Twin Fang 0.5s (5s without poison).
Karthus: Lay Wast 1s, Defile (how does your new system interact with such a spell?)
So instead of balancing spellvamp by its effectiveness on Vladimir and Akali, we balance it w.r.t. Cassio and Karthus instead. Furthermore, Akali for example, intended to function as a hypercarry, is completely balanced around having such huge sustain. Hence, even more balance work will be required. So I don't really see the benefit of your solution. It just creates a different set of problems, a different class of spellvamp users.
Beyond, I'm confused by your terminology. A "hyper mage" is something that currently doesn't exist in Riot's vocabulary. You want (some) sustained damage mages to be on par with hypercarries like Kog'Maw and Tristana? However, given that these mages are significant sources of CC by themselves, e.g., Anivia provides a huge slow field, Malzahar a suppression, Cassiopeia an AoE stun, etc., this seems unwise. Only mages that have very significant / low amounts of CC or give up large chunks of power in other areas can be considered for a hypercarry position.
For example, Akali is allowed to be a hypercarry and have substantial sustain because she directly risks her life in melee combat, having no hard CC by herself at all and only dealing single-target damage. Similarly, Karthus creates a zone of high pressure AoE damage by essentially suiciding. He's a low mobility mage with no CC besides a strong slow. In contrast, Orianna is a high pressure (lane bully), safe farming mage with a teamfight deciding AoE CC ultimate. The latter clearly doesn't need buffs when she's already constantly being played in LCS.
Having said that, let's once again discuss the fundamentals of Spellvamp:
What do you see as the current issues with the spellvamp system?
Primarily, I estimate you want mages to also utilize spellvamp, not just ressourcless / energy-based characters. You want spellvamp to be a more general stat to mages, similar as lifesteal is to marksman.
However, the question arises whether that is actually a valid goal. Should mages really have access to sustain? You yourself even said that burst mages in general should not. So we have to immediately restrict the set of viable champions to sustained damage mages.
On that specific champion pool, should the statistic be strong early or late game? Clearly, times and times again sustain has been determined to be a negative when introduced to the early game in too high of a quantity since it reverses player interaction (damage taken) and turns the lane play stale. Hence, spellvamp should be strong late game. This should create an opportunity for sustained damage mages to provide their output longer to a fight, similar to a marksman.
I hope I've correctly identified your position so far.
Next, what should we do about the above, while minimizing fallout? That's a really hard question which we shouldn't answer presumptuously. I believe we should first identify the core problems more throughly: Locate the set of champions that are currently utilizing spellvamp, the set of champions we want to support and the impact we expect it to have. Only then we are able to make an informed decision.