A corrupt rural sheriff must find a senator’s abducted daughter to bring a quick end to an FBI manhunt that could expose his ties to local drug trafficking.

Why do we root for him?

I think both Richiev and Valentin raise interesting points.

For me, one of the similarities between "Man on Fire" and "Silence of the Lambs" is that in both cases there is a strong emotional subtext/link between the protagonist and the victim. Both allude to the linkage in "Man of Fire", the bond that developed in the 1st Act between Creasy and Pita.

In "Silence of the Lambs", I interpreted the scenes of the trapped and terrified victim in the serial murderer's pit as connecting to the terrifying memories Clarice had of the lambs trapped and being slaughtered -- memories that Hannibal Lecter forced her to confront. (Which was sufficient dramatic justification for me for the scenes of the victim's entrapment and terror.)

In any event, I agree with and would amplify Valentin's observation that there needs to be a personal connection between the victim and the protagonist. Personal in the sense that the sheriff knows her or personal in the sense of a shared emotional conflict and/or psychic trauma.

In "Man on fire", the theme is different. It is about a relationship between a broken man and a lonely little girl. How she saves him emotionally. Denzel Washington is introduced as a wash up alcoholic. The little girl behaviour and incessant questions means that he can see a way back into humanity for him. There is love, trust and respect between the two characters. When she kidnapped, his motivation are personal. He goes back to being a killer, but this time he has a purpose and does not feel bad about it.
From the logline, the situation is completely different. Here, the sheriff does not know the kidnapped senator's daughter. He has had zero interaction with her before. So she can't be his internal motivation. He does not care about her, just about the concept of her. Hence the McGuffin.
One way to give him personal connection with the kidnapped victim would be to make her a surrogate. Maybe she remind him of his lost/killed daughter, but then what need to be shown is how he lost his daughter. Maybe he is a former FBI agent/police who got a kidnapped victim killed. Then again, we want to know how the previous kidnapping unfolded. In all cases, we would want to know why she is a surrogate, of whom she is surrogate and what happened to the original in order to understand him. It's not about the victim, it's about making things right this time according to the protagonist.