Would this solution to pedophilia work?

There is NO controversy. Pedophilia is a mental illness where the person is sexually attracted to pre-pubescent children. We do NOT criminalize having a mental illness. Thus, the mere presence of the illness is not and should not be criminal. If/when the mentally ill person ACTS on the mental illness, the ACT can then be criminalized. The crime, however, is sexual assault, molestation, rape, etc. NOT pedophilia. There is NO jurisdiction where "pedophilia" is listed in criminal code. What happens at the point of arrest for a crime acted out by a pedophile may be up for debate.