If I had been involved in a search for a missing child, I'm sure I'd hate this show, too.
I tend to stay away from shows/movies that will trigger me.
But if movie/tv producers/writers freaked out about everyone's triggers, all we'd see are reruns of Mister Rogers.
As for the Harbour character, I thought he was pretty genuine. Drunks aren't all assholes. In fact, most times, they're just sad. The violent ones stand out, but his character wasn't a violent drunk, he was a sad drunk, like many in the world.
And, he was a former boyfriend of the kid's mom. He'd be a sociopath not to want to help her find him. Alcoholic != sociopath. Usually, they're people who have the worst kind of food allergy, and when they seek help, it should be there for them to receive.
I only saw the first season and a half myself. I thought the first season was good, but I quickly grew tired of its tropes, so I'm not sure you're missing all that much.