The wikipedia article I linked to has some info in English on the different groups of perpetrators. Then there is the link to the article on a study which analysed cases of convicted child abusers in the court district of Stendal, in German. I am sorry that so many of my links are in German, and I couldn't translate much (or else I still wouldn't have finished the diary). I can sum up this one a bit, though, from the findings of the article that I took the picture on relationships between perpetrator and victim of: 20% of the perpetrators are strangers. In this group a significant (but unfortunately unspecified in the article) subset is mentally disabled/has an extremely low IQ: men who cannot have normal relationships with other adults, but are sexually attracted by them. They seek sex with children, who are intellectually their equals, as a substitute. This subset probably comes closest to the description you have in mind with your questions.  

80% of the cases are committed by persons who know the child well. They are close relations, who are in a position of power. The incest taboo is obviously working: stepfathers are more likely to sexually abuse children than biological fathers. A large portion of sexual abuse happens in dysfunctional families where a man batters and rapes wife and children. Nothing to do with paedophilia, this is violence against persons he treats as property. The motives of siblings, grandfathers and uncles fall into the same category: domination. Children are chosen as victims because they are helpless and cannot even judge the situation as wrong (many abusers tell their victims that all fathers/brothers/whatever do this to children, that it is normal). The satisfaction of the perpetrator is in the power relation. 42% of the perpetrators belong into this group, and I don't see paedophilia as a motivation here.

The last group (38%) are acquaintances of the child or its family. Neighbours, baby-sitters, teachers, priests belong here, too. Positions of power and trust. We can't be sure how many paedophiles seek these positions in order to find victims or how many perpetrators find themselves in these positions that they then exploit to exercise power over children without being particularly sexually attracted by them.  

So, plenty of motives for sexual abuse of children. Paedophilia is only one of them, and all evidence points at that it is the motivation for a minority of the cases. The focus on paedophilia does not help then.  The danger for children is in "exploitable power". That's for instance families with a violence problem. And that's teaching children that there are particular groups of people who can be trusted as a whole, like teachers or priests. Strategies to protect children are more likely to be successful if they come from the angle of "power" (and authority, trust) instead of "sex" (and body). That's why I insist so much on differentiating between paedophiles and child abusers.