First things first. After the fight with Goku Black and Zamasu, Trunks is almost killed in an explosion from the Time Ring after Goku sacrifices himself to hold Black in place while Piccolo (who has travelled to the future in place of Bulma) Fires a Special Beam Cannon at the two (sepcifically at the Time Ring). This explosion causes a ripple through time which turns Trunks into a baby as that was how old he was when real changes started happening to the main seen timeline during the Android saga. He is also propelled backwards to Planet Vegeta to the time just before Bardock was killed by Friezas blast. This event is what causes a time distortion and changes things so that Bardock is sent back through time instead of killed and he then goes on to fight Chilled in the special episode that we have probably all seen. Trunks being his future adult self in a babies body has a higher level than most at 10,000 which is kept that low due to him not being able to control it like Goku not being able to maintain SSJ3 in his child form in GT. He was feared by King Vegeta however Paragus claimed him as his own due to his power and planned to one day overthrow the throne using him. The hate that Broly (aka Trunks) has for Goku comes from the fact that the last thing he remembers before having his mind scrambled was fighting Goku who caused him lots of grief. As we know however it was actually Goku Black. He calls him Kakarot since there are traces of his memory left from how Vegeta would call him that. The first Broly movie show's Broly not killing Trunks. This is because he would not be able to survive if his past self was erased at that point. This is not something that he continuously knows but he does however feel an unseen force causing him to hold back against his younger self.
The same thing goes for Broly the second coming however this is where the Z Fighters first meet the Kaioshin. This is what begins their jorney in to meeting the hierarchy of the Multiverse and is a very significant time. At this point Beerus is asleep and Whis is looking over the Universe and sees Broly who he can detect has similar Ki to Trunks'. He realises that they are the same person by looking back through his timeline and decides that it would be too much work for Zeno to deal with a paradox being created of Broly killing himself and preventing himself from ever killing himself and being an infinite loop so Whis contacts Zeno who agrees and send Goku to earth for just long enough to help his family and Trunks in the final ki struggle against Broly. Zeno then uses the Super Dragon Balls to erase the memories of the other people involved as to preserve the timeline and keep it intact which is why there are no further mentions of Broly.
BioBroly does not influence the story in any way.
These events show why Zeno let Goku leave and go to the future as he knew that if he interfered it would have caused more problems for him. Without explicitly stating it, the button that Zeno gave to Goku was in case of an emergency in the future so that Zeno could go and correct it and allow it to play out as it should do.
At this current point, time has never flowed past the events of the Time Ring being destroyed therefore Zeno cannot predict the outcome of the Omniverse tournament. This is also the reason he agreed to let Goku bring a friend as he want's to see how everything plays out and if he made the correct decisions.
Feel free to ask some questions about this story. I can fill in blanks I may have missed