Kayle is your perfect stepping stone
At the core, you know what a scaling champion should be. We know because you created Kayle. She's not oppressive until later in the game. This is how a hypercarry should scale. Their early game should be utter shit and they scale into monsters later in the game.
What you do now is very important. Instead of buffing Kayle straight up, I suggest balancing the game around Kayle's concept. Bring back early/mid and late game champs. Balance it until where champions like Kayle get dukied on till they hit their spikes, but not made utterly useless until then. They should be easy to zone, but not so much that they explode at lvl 1 or from 1 rotation lvl 3.
Bring back poke/sustain and seige comps. I'm tired of aram death matches and nuke tactics. This could go so well...