Kog's big problem is reliability
Let's take a look at the other two true hyper-carries, shall we?
's Ult gives her a Big Moment with its stealth, movespeed, and hefty raw attack damage boost. Aside from popping it and tumbling into the middle of the enemy team, it can't really be wasted since Vayne has the tools she needs to to hunt down people who try to disengage from her. Even when her Ult is on cooldown, her Q and W provide her with formidable damage and her E is honestly pretty difficult to screw up. All in all, everything in her kit is very reliable. It does what you want it to do, when you need it to, with relatively little downtime.
's Ult likewise gives him a Big Moment due to its attack damage boost and piercing shots. His Q allows him to position largely with impunity to line it up and his Q resets allow him to keep his boosted IAS over the course of a battle. Flask is nearly impossible to miss, Contaminate serves to amplify his AD, and his Passive adds a nice little chunk of true damage. When his Ult is down, QWE with a few auto-attacks tossed in makes for a decent chunking tool. Again, his kit is very reliable.
And now
. His Big Moment is his W, which boasts 6 seconds of uptime and 7 seconds of downtime. Thirteen seconds is often enough time for a team fight to be well and over, so he typically won't get to pop it for a second helping. When Bio-Arcane is down, nothing Kog has is reliable. He's stuck with low attack range (for an ADC) and a gimpy item build that needs his W to function. His passive can be dodged. His Q can be dodged. His E can be dodged. His Ult can be dodged.
If Kog is going to work as League's immobile long range sustained damage Marksmen, which by all accounts is what Riot wants Kog to do, then his kit must have more reliability. By the time the team-fight stage rolls around, Kog should be able to always have Bio-Arcane Barrage available for use if he's to function, because if he doesn't have it then he has no way of defending himself, let alone fight back with any degree of success.
Let's make this really simple. Reduce the cooldown on Kog's W to 12/9/6/3/0 seconds. Still enough of a cooldown early game to prevent him from having it up constantly (since Riot insists that Kog needs to be awful in a meta that strongly favors early/mid game ADCs) but by the time skirmishes and team fights start happening, he'll be able to fill his fantasy reliably.