Yeah I think maybe I was testing the wrong side (how much a teensy bit of horror would turn you off). There's a good deal of really good positive, lighthearted slice of life anime, and plenty of hero stuff for kids. I don't watch much of it though. :(
Sports related positive hero stuff: Haikyuu, Cross Game (Amazingly good, moderate relationship stuff), Hajime no Ippo (Seriously just fantastic)
HERRRROOOOOOOOOOOOOOOOOOOOOO! : One-Punch Man (Amazing)
Hero plus Iron Chef (seriously): Food Wars! Shokugeki no Soma (Really good)
Slice of life plus people playing piano: Nodame Cantabile
More serious, but awesome: Dennou Coil (scary for kids), Planetes, The Melancholy of Haruhi Suzumiya, Samurai Champloo (very unique, can't remember but there might be a little horror), Haibane Renmei
Sorry, they aren't ordered in any meaningful way. Just look them up and pick one that looks good to you. They are all fantastic in their own way.
Things I've heard are good but haven't watched myself: Ano Hi Mita Hana no Namae o Bokutachi wa Mada Shiranai, Clannad/Clannad: After Story, GTO, Gintama'