WEll, having had some experience with this, I can say your premise generally is pretty sound. Sure, a random comment by kid to someone else could wind up getting dad investigated, or a comment by mom. Or, if they are going through a custody case, and it's contentious, there may well be a referral to a psychologist for a full evaluation of both parents. You bet parents will make accusations in those evaluations that psychologists are required to follow up on (report, not investigate). That's why many psychologists (me) hate doing those. Both parents are trying to look good and make the other look like a loser, and in the process, the kid can really lose.
Psychologists do not have opinions about what might have happened in the past (well we have lots of opinions but we wouldn't be able to say, "He did it, for sure, the tests prove it!" ) A psychosexual evaluation can help determine how well they'll follow treatment, chances for recidivism, how much risk they pose the community and so on. So PSEs are typically done after someone has admitted their crime.
A privately retained psychologist is going to be covered by confidentiality (not so a court mandated psychologist, who is, say, performing an assessment where the client is actually the court or CPS, confidentiality is very different in those cases.)
Sure, an accusation could result in removal of kid or offender from the home. But, to go forward, there is going to have to be some evidence other than "mom said." Otherwise the case gets closed as "unsubstantiated." If mom discussed this with a neighbor, and the neighbor suddenly comes forward and says, "He molested my little Katie too!!!" that would get attention (as happened here, all proved to be false.) In the Good Mother, a good example, they do some ill advised stuff (I think he showers with the girl) but, they did do it, no question about false accusations, just about whether or not what they did was abuse, as I remember.
But yes, false accusations do indeed happen and not as rarely as we would hope. As I said, we have one happen in 2007? maybe later, with multiple accusations, looked bad for the guy, but in the end, he had a very good lawyer, and demonstrated all the accusations were false, the parents just got on the bandwagon. Didn't really help the guy though. Sure, he got off but not like he'll ever work in the field again. And he did have to move. Another individual was also exonerated by the courts but his ex still got a restraining order, and prevented him from seeing his kid, who now hates him (no contact for ten years.)