Okay, taking all that I know and everyone else's comments into consideration, I'm going to drop some knowledge in this discussion. It might get kind of weird so bear with me.
We'll start with characters being born with/not born with tails. All full blooded Saiyans from Universe 7 are born with tails as we know. I believe it might have been in the Daizenshuu guides, but I remember seeing information that says in regards to half-breeds getting tails or not, it seems to be a recessive trait. My own two cents in the matter is that more of the kids should have had tails, it shouldn't have been just Gohan and that's it. Non-canonically, I like how MasakoX gave Ranch a tail seeing as she's the daughter of Raditz and Launch, but their other two kids Daikon and Mooli do not.
As for information regarding why the tails sometimes grow back, well it's more spotty but Daizenshuu states that they'll grow back suddenly if the Saiyan is in danger or great peril, kind of like a need to protect themselves with their Saiya power, that gets relative to Broly but more on that later. Re-growing of the tail in an emergency was clearly shown during the battle with Vegeta on Earth where Gohan's tail suddenly regenerated after Vegeta beat the crap out of him. No solid information exists why Vegeta never regrew his tail. Some surmise because it might be due to him being an adult. Only kid Goku and Gohan have been shown re-growing their tails. If my memory serves correctly Goku didn't regrow his tail because Kami did something to remove it and prevent it from growing back, but exactly what is never explicitly stated. It also could be due to age because it was around the time he undertook training with him and Goku was nearing his later teen years at that point.
My opinion on Broly using Wrathful form as some people call it could be this need for power. It could also be relative to him being the Legendary Super Saiyan. It could also be some wink and nod to Super Saiyan 4, a reintroduction of the form, hopefully! It's just speculation on my part.
Dragon Ball GT is a mess of plot holes in regards to Saiyan tails and Super Saiyan 4. Why does Goku need his tail to transform into a Golden Great Ape but Baby Vegeta just gets hit with science and transforms? Also, after attaining Super Saiyan 4, why does he not have a tail when he goes back to base form? It comes and goes to the same place that their clothes do when they transform into Super Saiyan 4. It's also the same place that Optimus Prime's trailer goes to/comes from.