I switched to a thumbless grip a.k.a. false grip a.k.a. suicide grip a.k.a. monkey grip for any barbell press after watching this video:
Basically, Tate recommends setting up and gripping the bar tightly/securely with the thumb around and then, just before the rep starts, flipping the thumb into thumbless position so you still have tension from the hand to the wrist to the forearm and up through the arm, compared to a relatively-less-firm grip from “just” grabbing the bar thumblessly.
EDIT: Tate does also discuss the “thumb-around:pec activation, thumbless:triceps activation” that Spidey brought up. Interesting.