Goku wasn’t very powerful to begin with. He got his ass effortlessly handed to him by Raditz. It’s possible that Vegeta had more potential than his father, but then we also don’t know much of anything about King Vegeta to say that he was weak. All we do know is that he was killed by Frieza, and to be fair Frieza is a prodigy with unimaginable growth potential.
There are really only two things that make Goku and Vegeta different from Bardock and King Vegeta: Goku and Vegeta have both died more than once, and they both lost their tails.
King Vegeta and Bardock never had the luxury of having people wish them back with the Dragon Balls. And as they were both vicious space pirates under Frieza’s employ who have wiped out countless civilizations in their lifetime, it’s safe to assume that upon their deaths they were not granted the honor of retaining their physical bodies. This means that they could never be wished back, as their souls have already been reincarnated. Had they been wished back, as Goku and Vegeta have been on several occasions, they would live to fight/train another day, and possibly would have become every bit as powerful as their sons did. Or even if they had the honor of keeping their physical bodies in the afterlife, they possibly could have trained with the Kais and gained greater power like Goku did.
In regards to the latter difference, the absence of Vegeta’s and Goku’s tails, there is a pretty obvious correlation between the power a Saiyan can gain, and the presence or absence of their tails. Vegeta is the perfect example of this. Vegeta had been wiping out planets and civilizations since he was a child. And while he was considered fairly powerful, his power level was roughly only 18,000. Within 20+ years of training and fighting, Vegeta was still significantly weaker than a lot of Frieza’s best men (Zarbon, Dodoria, the Ginyu Force), and he still had his tail. Fast forward to his time on Namek after having his tail cut off on Earth, Vegeta managed to gain power well into the hundred thousands before being killed by Frieza. That is a massive jump, to go from 18k in 20 years, to over 100k in way less than a year’s time. And the only difference between 18k Vegeta and 100k+ Vegeta: 100k+ Vegeta no longer had a tail.
With these couple of factors in consideration, that being the presence of a tail and the longer lifespan, there’s no reason to think that Bardock and King Vegeta could not have reasonably become as powerful as Goku and Vegeta, or at the very least achieved Super Saiyan and power comparable to our tailless protagonists.