Carter wasn't recruited to come to Eureka. In the pilot episode, Jack explains his presence in the town. He was taking his daughter to L.A. (she ran away from her mother) when they encountered themselves traveling out of town, which led to an accident, which led to them stopping in Eureka (to seek help) instead of simply passing by.
Carter: Well, I was transporting this one to LA when we had a little bit of an
Carter: I mean, one minute I'm driving my delinquent daughter back to Los
Angeles, and the next I'm in the middle of the freaking Twilight Zone!
At the start of the show, Carter was a fugitive-chasing US Marshall, not a town sheriff. After the town's existing Sheriff, Bill Cobb was incapacitated, Marshall Carter was recommended for the position as his replacement.
Beverly: Actually, we're here on behalf of the Major. He was very impressed.
Carter: You mean Colonel Colonel Briggs.
Beverly: No, Major William Cobb, Army Intelligence. He said you two worked together recently.
Carter: Oh, you mean the Sheriff?
Beverly: With his recommendation, your security classification and pay level will increase, effective immediately. Marshal. Sir.
Carter: I don't understand.
Beverly: Relax, Carter. It's a promotion.
As to why he was chosen for the role, it most certainly wasn't for his genius (his IQ is repeatedly stated to be determinedly average) but for his tenacity, insight and people skills. This allows him to be the "everyman" in a town of geniuses.