No, that's not true. He does perform a lot worse late game but of course he's still a carry and literally all he has is damage (no utility or CC is somewhat rare on champions). Check champion.gg or leagueofgraphs.com. A lot of that is based on the fact that his spells are not quite as meaningful late game when autos chunk for more on typical ADC builds. However, Corki doesn't have great AA steroids and he also can't itemize for penetration well because his autos are split damage. Thus, for the biggest damage source late game (autos), you're basically getting half value from a penetration item. He also just has a bursty kit rather than insane DPS but that falls off a bit in teamfights if you get proper peel and itemization. Corki excels in skirmish scenarios.
His winrate is highest early and goes linearly down the longer the game continues. He was probably just fed or you guys didn't build correctly. Every champion gets stronger late game and it's really about RELATIVE strength assuming equal gold. Corki is definitely good both mid and bot but he isn't overwhelming. Just pay attention to his spikes and Package sound. The best advice I can offer is to play him yourself and your opponents will eventually teach you how to counter him.
BTW: I wouldn't call a 26 CD 100 mana 650 range dash a "great escape." If you're talking about his enhanced W, that is pretty amazing but only every 5 minutes and he doesn't have long to use it.