QIf an opponent only has a Heavenly Aegis, which has Aura and "This follower can't be destroyed by abilities," on their field, can this follower's Fanfare ability cause them to bury it?

AYes. Aura prevents the card from being selected by enemy abilities, but in this case Heavenly Aegis is not being selected by an ability. Also, burying a card isn't the same as destroying it.