Beerus would've dealt with Boo and Moro due to their destructive nature and the threat the former placed on Shin. I don't believe they would've dealt with Freeza, however, as he didn't present a destructive threat of a universal scale and the neutral and measured roles they have would mean they most likely couldn't interfere with matters they personally disagreed with. That said, the series goes the same up until the Boo Arc, which has either no or limited conflict. Beerus still goes to Earth seeking SSG and Goku eventually resorts to the Dragon Balls, with things going the same as in canon for BoG and FnF.
Universe 6/7 tournament doesn't happen due to Beerus knowing it'd be an abuse of his power. He talks Champa out of using the Super Dragon Balls by threatening to go to the Daishinkan. Future Beerus being alive means Black and Zamasu don't go to Trunks' timeline and instead go to another universe or U7 timeline. One of two things then happen:
1. Zamasu goes undisturbed and succeeds in the Zero Mortal Plan. Zen-Oh erases all universes and timelines.
2. Zamasu is stopped by Daishinkan or Zen-Oh. The ToP doesn't happen due to Goku not meeting Zen-Oh. Broly Arc still occurs as normal. At some point in the future, Zen-Oh erases all but 4 of the universes.