Carroll might have decided to go for corruption only after having repeatedly failed the other way.
Or he might have developed good corruption skills only after a numer of years (corruption is indeed some kind of talent, some people are better than others, some are naturals, and some are just not capable of doing it).
Or he might have had the right mixture of corruption-kid talent-performances just after a number of attempts.
Mind you tho: I'm not saying I think Carroll bought judges. I'm just pointing that the fact he only won OG this late in his coach career is no indication he didn't.