Yes it does, he killed his wife plus his son can't fight him so it's his job to take over.
In the Buu arc he said Goku was better, not that he won't try to catch up. He just doesn't hate him and want him dead like before. There are better writers here than me so if I try to improve my writing does that mean I'm going against what I said about them being better ?
He saved Goku from #19. Piccolo even said that he might've been stronger. He also helped Goku get his body back from Ginyu, pulled Buu out of Buu and came up with the methoed of which Goku used to kill him.
I'm getting a little tired of the back-and-forth so won't try to carry it on for much longer, but here goes.
The Bulma that died wasn't "his" Bulma, it was Future Trunks'. And getting back to my original point, I prefer the anime's pre-Zeno ending where Trunks does in fact defeat Zamasu. Cards on the table, I like the Spirit Sword, I don't care about the logical shenangians, I felt it was a satisfying climax that tied into the major themes of the arc and allowed Trunks to re-earn his role as the Bringer of Hope to his world instead of relying on his dad to do it for him. Vegeta's role in that story was to be to Trunks what Goku was to Gohan in the Cell arc. They even did a Father-Son Galick Gun. If Vegeta ever had to get a win against a main villain, it shouldn't have been in that arc.
I felt like Vegeta was one of the best characters throughout most of Super, it's only very recently, like post Bra-birth where he seems to have regressed into being obsessed with surpassing Goku again. Before, he seemed willing to accept he was at his limits with Super Saiyan Blue and wanted to pass the torch on to folks like Cabba and Future Trunks. Ironically he's gone through the same regression Goku did after coming back in the Buu saga; he's stopped caring about the next generation surpassing him and back to being obsessed with his own growth. And that means he's back to being aloof and going on and on about needing to be "the best" again. It's boring compared to the interesting characterisation we had up to now.
All the examples you put of Vegeta being better are actually just examples where Goku was weakened by another influence. Him being stronger than Goku when he showed up as Super Saiyan for the first time is much of a muchness since both of them were well above #19. I don't know if you've watched the ep where they went inside Buu recently, but Goku was carrying Vegeta through most of it. Vegeta was only able to pull Fat Buu out because Super Buu's attention was on Goku and he didn't notice Vegeta had regained consciousness and made it over to Buu's pod.