"I have seen the "GLers cannot be Molesters" argument before. It is false."

I think we have here two different sets of definitions. You seem to be defining Girl Lover as a perfect synonym for girl-attracted pedophile, whereas many use the term Girl Lover to define that sub-set of girl-attracted pedophiles who do not merely lust after girls but love them and treat them with respect - which is to say, girl-attracted pedophiles who are not molesters because molestation is not respectful.

I think everyone agrees that having a pedophilic sexual orientation does not make a person wise and loving in itself, and that pedophiles can be molesters (though it seems to be slightly less common than among the general population), but for many the claim "GLers cannot be Molesters" is true by definition, unless one possibly excludes rare instances of accidental molestation.