Are my siblings and I secretly competing for our dad's approval and not admitting it?

Almost certainly yes, and admitting it openly is usually the only way to defuse the competition. Kris Kluver, in The Dysfunctional Family Office, has Gail Mitchell observe her brother Steve and notice that even though Steve says he doesn't want the company, his eyes show that he wanted his father to choose him. The hidden competition is one of the most universal patterns in UHNW sibling dynamics. Each sibling is reading the others through the lens of who Dad favored, even when nobody admits the framing. The competition can run for decades, distorting every family interaction. The fix is to name it. One sibling, in a private conversation with the others, says out loud what everyone already feels. The naming usually breaks the spell. Pretending it isn't happening just keeps it alive.