My take is quite different. EVERYONE in Broadchurch is at least nearly-criminally incompetent.

"Ooh, I'm an investigative detective in a homicide. I think I'll forget myself and beat up somebody in lockup!"

"What's that, evidence? I think I'll withhold it for minor personal reasons."

"Hey, there's a pedophile investigation going on. I think I'll lie about my 'alone time' with a teenage boy to EVERYONE, just to avoid arousing suspicion..."

Tennant's advantage is that, in season one, he's not emotionally tied up in this completely tangled small town. He's got some professional competencies over Miller, but not many.