First, you need to separate clinically diagnosed pedophilia from legal terminology. Legally, pedophiles can mean anyone who ever once engaged in any sex-related crime involving legally under age persons.
Clinically, actual actions are not required, just obsessive thoughts and fantasies, and the children need to be pre-pubescent. A teacher who is otherwise normal but winds up attracted to and having sex with one particular 14 year old student of theirs might be called a pedophile by the law, but is not a clinical pedophile.
Clinically speaking,
Pedophilia is a type of paraphilia—a category of recognized mental disorders defined by unusual fantasies, urges, or behaviors that are recurrent and sexually arousing. In order for pedophilia to be diagnosed clinically these thoughts or behaviours must be present for at least six months and must cause distress to the affected individual or impairment of the individual’s ability to function socially or occupationally. A clinical diagnosis of pedophilia also requires that the affected individual be at least 16 years of age and at least 5 years older than the child (or children) at the centre of the individual’s sexual fantasies.
Note the distress and impairment features of the obsession, which indicate it is self-destructive nature and thus a disorder and an illness. These are not just people that broke the law by taking an opportunity that presented itself to have sex with a minor. They are people whose sexual thoughts and arousal center primarily and often exclusively upon pre-pubescent kids. Many hate themselves for it, and/or never act upon it but cannot stop the thoughts and arousal. Sounds like an illness to me.
All thoughts and actions are caused by the brain, but whether it is a brain-based illness is another matter and assumes that the roots are biological and that experiences play little role.
The wiki on it suggests a emerging evidence of root neurological factors, including brain structure differences at birth and early childhood head injuries that predict the disorder. And other research suggesting that being abused oneself is not a cause of the mental disorder, but can be a trigger for actually acting out the fantasies and committing criminal acts.
There will be strong social resistance to thinking of it as a mental disorder, especially a brain based one, because we so strongly hate the actions that we want to hate those who commit them, and a biologically based disorder throws a wrench into such punishment craving judgmental anger.