I don't think Kog can be balanced AND healthy as a hyper-carry

Take a look at who he competes with in the hyper-carry department.

Three of the most picked champions in the game and a niche Marksmen/Assassin who can ace your team if he manages to get fed early. If Kog were strong enough to be as popular as the three gals above, he would have to be flat out broken. And if he's kept at 'balanced' instead, there will really be no compelling reason to pick him when you can build a 'Protect the Marksmen' comp around someone else who can be just as frightening with the right team.

So at this point you're probably wondering 'Well, if Kog can't be a hyper-carry then what the hell CAN he be?'

Well, my hypothetical inquirer, let me tell you.

Kog had two big things going for him before the remake. Crazy base damage on his W/R and potent poke with W/R/Sheen.

He lost both of those with the remake. But what if we gave Kog that intense base damage back again? What would that look like?

Well... we would have a super-immobile champion with great base damage, heavy sustained damage, crappy CC, virtually no utility, and arguably the most powerful global taunt next to Teemo. A turret champion. Or a tri-pod machine gun, if you want to go that way. We would have, effectively, a ranged Juggernaut.

And why not?

Graves is already a Marksmen/Fighter.

Why can't Kog be a Marksmen/Juggernaut? He certainly doesn't look terribly fragile when compared with someone like Jinx or Ezreal. And giving him durability as a second lever in addition to 'all the damage' would probably make balancing him a little more workable. There are a ton of ways this could be accomplished, everything from tweaking his raw base/scaling stats to give him a new Innate that lets him go toe-to-toe with angry divers without popping instantly.

I say take the remake one step further.

Let him loose on the Rift so everyone can learn first-hand that TanKoggy gives no fucks.