Seems like you are arguing a technicality while avoiding the substance of the question.

And, of course it is fine to argue a technicality in a legal discussion when discussing statutory language.

But, "strict liability" is often used as a catchall term (as opposed to a statutory term) to casually describe a class of statutes that remove or lower the mens rea (mental state) standard, while still requiring the actus rea (act performed) to be proven by the prosecution.

You argue that "statutory rape is not strict liability" because "the State still need[s] to prove...intercourse," (e.g., must prove the act while still setting aside the mental state or awareness of the accused regarding the illegality of the act...I'm not sure it is much of a defense for a guy to say he didn't know he was having intercourse). You could just as easily use that logic to argue that traffic violations are not strict liability because the State still needs to prove you were speeding, while setting aside any awareness that you were driving a car. So, it seems a technical distinction without effect, in which case...

...The question remains: Why should there be a lower standard of proof for rape than for murder?

The answer probably hangs on the amount of punishment which attaches to the crime.

But, perhaps there should be a fall-back strict liability homicide law that replaces the "not guilty by reason of insanity" statute, and the guilty person could still be mandated to receive mental help in prison. This would make it easier to imprison people when there is no question that they performed the murder, and the only question is the mental state of the perpetrator, which is difficult to prove, to say the least...almost impossible to prove.