Read the whole thread. Some great ideas. I agree with most of what Rid Prime said, and probably all of what Kento said.
I would make the power scaling normal. So, by the end of DBZ, Goku and Gohan would be destroy a quarter of a galaxy with just a normal Ki blast. I would make fights have to occur in a different dimension with Piccalo always transporting them.
I would still make SSJ4, but I would change the look of it completely. As well as SSJ3. The hair on SSJ3 and the missing eyebrows are stupid looking Whoopi Goldberg wanna bes.
I woulda made the SSJ4 the actual LSSJ and the whole thing about the LSSJ would be changed.
Vegeta would defnitely reach LSSJ, but far after Goku did. He would have to struggle to get there, just like he did for SSJ. VEGETA would have been the one to kill Freiza, not Goku.
Also, in the Saiyan Arc, I would have Goku hand Vegeta his a$$, and not make it nearly as even. Vegeta would NOT turn great Ape as he would be too arrogant to try that. This would make Vegeta have an epiphany of, "what the hell have I been doing all these years." Of course, Goku wil ls spare his life, and Vegeta will grow a soft spot for the Z warriors after that...however, he will still hold a grudge against Goku, a low class warrior, defeating him.
Goku's father would still be alive. Instead of planet Vegeta getting blown to bits, it would have been all of royalty (minus vegeta, of course. Vegeta would witness this, but pretend he didn't know, and work under Freiza his whole life until the Saiyan wrc) and all of the upper class noble saiyans...cause they would confront Freiza about treating the saiayns poorly and then Freiza would pwn them dead. Then, all the rest of the saiyans would be treated like dogs and low level henchmen, creating the hate. And starting up the events that lead to Freiza saga.
Of course, Goku would meet his father after the Saiyan saga, and Bardock would be a harsh, tough, rough, father, but still have a soft spot for Goku and still show it, but subtley. Bardock's backstory would still be mostly th same, but diverge when he tries to escape to find his wife (once he found out Frieza was coming...Bardock wouldn't actually try to lead the attack, King Vegeta would.)
What would set Vegeta off to go SSJ is something I haven't thought of, yet. Cooler would exist, and Goku would fight Cooler and go SSJ against Cooler. Goku would go SSJ against him because Cooler would execute Bardock, right in front of Goku.
NO SSJ kid trunks and kid Goten. They all would have to reach SSJ when they were in their 20s. The eariliest SSJ would still be Gohan (at 13?) It would stay that way.
Broly would be an entire saga after the Freiza and Cooler Saga. Broly would NOT destroy an entire galaxy, however. His form would still be USSJ with the speed of an ASSJ (like Vegeta.) Vegeta would beat down Broly, after coming out of the HBTC. He would have to work for it. Goku goes in, thinking Vegeta won't make it. HBTC training takes 14 days, instead of 1 day. The saga ends with Vegeta finally letting go of his pride, and accepting power from a namek and a half-breed saiyan (future trunks), which allows him to not kill, but knock out Broly. In this arc, the ultimate foe is Broly's father, but Broly still breaks free of the mind control device (but the mind control device never limits Broly's power.) Vegeta finds out about the mind control, and confronts Paragus about Paragus' plans to take over the now successful saiyan empire with what Paragus thought was the only SSJ. A fight comences and in so doing, Broly walks in on the fight, and witnesses Vegeta kill Paragus for his crimes against the Saiyan throne. Broly challenges Vegeta to an honor duel that will be held in 3 months. (Vegeta doesn't yet know that Broly has the USSJ form with the speed of a ASSJ, however, he does know that Broly is stronger than a regular SSJ, based on some sort of polot point I haven't thought of yet. So, they all decide the HBTC is their best bet.
Of course, Vegeta eventually wins (as I said before) with a similar "lend me your power" type of scenario. After Broly is conscious, Vegeta, Piccolo, and Gohan all prove to Broli, with the help of Bulma figuring out the device, that Broly was being controlled. Broly becomes part of the Z family. Goku and Gohan were in their training in the HBTC, during this time, and still have 13 days levft to train.
So, when Goku and Gohan come out, they step out into....
Dr. Gero would use that data to make his androids. When Goku comes out of the HBTC, all of the events for the android saga have taken place with Vegeta getting beat down and Broly taking a fall to Android 16. Cell has come along and setup his tournie and absorbed the others. Just as before, Gohan would still do his SSJ2 thing against Cell.
That's all I've got so far for my re-write.
Also, I would change the dynamic of father-son relationships for Saiyans. Still, a son would kill his father, but only when the father had reached old age or the point of not being able to fight anymore. The son would then have a fight with his father, and hopefully, kill his father. This would be considered the most honorable way for a Saiyan to die, other than in battle.
Oh, and, somehwere in here, King Cold comes to Earth for revenge with Future Trunks saving everyone again. (At this point, Goku still takes a while to get back to Earth, as well, as he decides to go off and train. We can pretend that Cooler is the older more powerful brother, but Freiza was the one more interested in ruling, so he was the one who ruled.)
So, to sum up:
Bardock's alive.
Freiza only killed the Royalty and noble saiyans.
Goku beats down Vegeta.
Vegeta and Goku go SSJ against Frieza and Cooler, respectively. They defeat them. Vegeta becomes the Saiyan King, and sets new policies that have the Saiyans as mercanaries, but also more industrious and intelligent.
Broly is part of the main story and Veget beats him, and tells him to submit to his rule. Broly is brought around by Bulma proving that Paragus was mind controlling him and Vegeta was correct in executing him for Paragus Forcing Broly to murder all of those people.
The Android and Cell Saga finish and start, respectively, while Goku and Gohan are training to fight Broly, whom they thought Vegeta would never beat.
Gohan goes SSJ2, and beats Perfect Cell, just as before.
From there, I'm all out.