Beerus would still visit Earth as that was as far as Whis' intel reached when it came to the Saiyans and events would play out exactly the same until Shenron is summoned. In wishing for the Legendary Super Saiyan to be brought before them, Broly would appear before Beerus but be unable to use much more than his Ikari state for survival. In pushing him to unleash all his anger, Beerus would have no choice but to admit he was the one who convinced Freeza to destroy Planet Vegeta and though having little connection to them, what it meant for his father due to his life there and how much Broly cares for him would have the desired affect. Broly would transform and gradually increase his power until reaching the peak of his LSSJ form. Beerus would barely win with his experience and upon finding the Saiyan who could match him, would choose to take Broly on as his successor.
Vegeta still trains with Whis due to Bulma's bribe and Goku begs for training too. When Freeza shows up, Broly kills him due to what his actions did for Saiyan society. When the U6 tournament happens, Beerus adds Broly instead of Monaka, though being saved for a last resort, he doesn't do much but oneshot Hit. His role in the FT Arc is minimal considering how little Beerus did. Broly replaces Freeza in the ToP and going by manga-scaling, his presence causes Jiren to become a non-threat pretty quickly. During the Moro Arc, it's unlikely Broly wouldn't take notice of Goku and Vegeta in Whis' crystal ball, leading to he or Beerus quickly resolving the conflict due to it concerning Dai Kaioshin.
In other words, Super just becomes Broly ex Machina.
