After a few games on Rift, I think I've identified why Koggy is so great on Dominion
I play Dominion, mostly, and play almost nothing but Kog. But the recent influx of account farming bots (seriously, Riot, do something about it) has made getting a fair game nearly impossible. So I decided to suck it up and give Rift a shot. After a few successes and a few failures, I think I've managed to get a good feel for why there's such a massive disconnect between Koggy's performance on the two maps.
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Early game doesn't exist. You start at level 3 and with enough gold to rock a Recurve Bow and a Ruby Crystal, which is enough to make Koggy a pretty significant threat right from the gate. That means the enemy team has less time to kick Koggy around like a pinata filled with gold and vomit.
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There's no laning phase to worry about. Whether you go top or bot, you can just play conservatively and snag Rageblade in a hurry. Once you've got it Kog becomes frightening.
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Skirmishes are smaller. Because most fights typically only have two or three people in them, Kog's short W duration doesn't matter as much. He can obliterate at least two targets before his W craps out on him. Over on Rift, larger team fights just means he's burned through the front line and now he has no way to deal with everyone else.
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The map takes care of his mana issues. Between the map's ambient mana regen, Health Shrines, and the fast recall/small size, Kog can easily manage his mana despite lobbing three or four Ults in every single encounter.
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Death isn't as crippling. Koggy is the leader among ADCs for most deaths per game (probably not the best title to hold). And on a map that snowballs like crazy, where one death late game can mean you've either lost or you're about to, that's a pretty big deal. Over on Dominion, Koggy can stockpile his revives for late game where he's the strongest in case he gets caught out to keep up the pressure.
Basically, Kog is great on Dominion whilst being a useless sack of shit over on Rift. The map covers several of his crippling weaknesses, though not all of them. He's still immobile. He still has no way of dealing with champions packing good disengage to waste his W. He still has no way of surviving a fight aside from his range and his damage. OnHit defenses (block and return) are still doubly effective against him. But he can compensate for a few weaknesses, and that seems to be enough.
Obviously nothing can be done about the death problem, but I suspect if Riot patches up one or two of the other weaknesses (mana issues and W duration would likely be the most impactful) I suspect Koggy could really shine on Rift as well.