Chekov wasn't intellectually stupid he was just an intelligent idiot, mathematically smart, capable of retaining facts, but a ludicrously poor judge of character. Making him a bridge tactical officer seemed OK to me; making him head of security seemed like a terrible idea (did he ever win a fight in TOS?), an opinion that was vindicated by every terrible security decision he made in STII, STIII, STIV, STV, and STVI.
NuChekov became a navigation expert, a transporter expert, an engineer, and then a chief engineer (all within a year of graduation) largely to give the character something to do but just because they both have a Russian accent and the same name does not make them the same character. They are absolutely, I repeat ABSOLUTELY, different characters.
I'd go as far as to say that Scotty, Uhura, and Spock are almost entirely different too. To his very great credit Pine manages to give you glimpses of Shatner's Kirk mixed in with the bad boy but it's only really McCoy and Sulu that bear a reasonable resemblance to the originals. Even if you subscribe to the notion that Sulu Prime was entirely straight, Cho's Sulu still has more in common with Takei's character than Yelchin's Chekov has with Koenig's Chekov.
That's not to say the new versions are not fun or likeable (although Pegg's Scotty is awful IMO). They are just different characters.