The ship travelling full quadrants of the universe is simply not possible. For context Goku's ship uses Saiyan technology for its speed, which comes with some problems.
Vegeta and Nappa's pods take nearly a year to travel from one unknown part of the universe to Earth. Realistically, if the pods can travel between quadrants in just six days, it would take far less time. It's never implied that Earth's technology makes it even faster,
in fact Earth technology at its peak is far inferior speed-wise,
heck the entire reason Saiyan technology is used is because its much faster. Both planets can also just reside in the closer part of their separate quadrants, which doesn't bring about the previous contradictions. We should just assume Namek is in another galaxy, due to them being in
separate quadrants that have their own sets of galaxies and whatnot, and then use the distance between Earth and the nearest galaxy (Andromeda) as a lowball.
As for why Vegeta's spaceship got there at the same time as the humans: Despite even leaving Earth first, he didn't even go straight to Namek. He first travels to one of Freeza's planets which took 18 days, healed there for 22 days, and then travelled to Namek from the last planet which took 5 days. The distance between these planets with Freezas is unknown, all it implies is, basing off of Goku's spaceship, is that the distance between Freeza's planet and Earth is far larger than the one between Namek and Earth, and the distance between Freeza and Namek is even shorter that Namek to Earth
which is logically possible. So this shouldn't be a contradiction at all, and that Vegeta's ship and Kamis aren't comparable at all even without accounting for Goku's ship, due to the different starting locations and distances to cover.
As for if Dr. Brief's could have made the ship faster:
It's already established that Earth technology, even accounting for what he could do, would be infinitesimal in speed. Nothing in the manga even implies that he could have made the ship even faster or anything like that, all he did was add on to it with a gravity room and made it bigger.