Summary- This is just a simple one-shot theory of why the saiyans were incredibly weak in the universe.
Think about the saiyans in Dragon ball Z for a moment. They were mercenaries that were able to wipe whole planets out and sell them in the Planet Trade Organization. But why were they so weak when they had various super saiyan modes and Zenkai boost?
First let's look at Saiyan Biology- They had the ability to turn into a Giant Ape at the sight of a full moon but very few saiyans could control it. They had the ability to heal after almost dying from battle very quickly and even developed a power boost from it aka Zenkai. They were also strong enough that they could unlock various modes of super saiyan that would double and triple their base power. Last they evolved to the point that they could fight for very long years and wouldn't grow old (but I doubt very many saiyans lived to be 40 or old but I'll get back to this later). Also their planet was 10xs Earth gravity so on Earth they were already at an advantage as soon as they stepped on our planet. Yet they were mocked by the aliens of Freiza's crew and weak compared to all the various aliens like Zarbon, Dodoria, and the Ginyu Force.
Now let's look at Human Biology- We don't have the ability to turn into Giant apes, it takes more time for us to heal and we don't get a giant power boost from near death exposures instead we generally get weaker. We don't have various super saiyan modes and we grow old very quickly. Also if you look at the few human warriors in Dragon ball Z they were generally weak compared to aliens and would die very quickly if put in a fight with them. Yet Humans survived Freiza but the saiyans didn't…Why?
The answer is Saiyan culture vs Human culture,
Saiyans believed you were strong if you fought and won many battles. Their training only consisted of fighting a lot of enemies. Bardock who was killed with the other saiyans and the planet by one big energy blast believed himself to be a "mighty warrior" solely because he was in a lot of fights. Also saiyans didn't strategize they mostly relied on "natural instincts" in fight.
Now look at human culture, we are weak compared to other races in the universe yet we developed training techniques that would help push our bodies to the next level. This training would also help us push our bodies to the next level so we could fight at an older age for example Master Roshi. Master Roshi was weak in comparison to Freiza but if you put him up against another human who was say 30 he would win despite being very old. Master Roshi also came up with various fighting techniques like the Kamehameha and the drunken fist style. Also we are very ingenuitive; we created things like the Gravity Chamber to help us train our bodies in various stages of gravity. So despite being laughably weak we are able to improve our race with various fighting techniques and train our butts off to accomplish new levels.
Now if you get a person with saiyan biology + human culture you get a warrior that became the strongest in the Universe hence Goku. He was able to use human training techniques that pushed his body to the limit and reap the benefits of a saiyan biology to become the strongest warrior in the universe.
When Vegeta was a kid his training consisted of being putting in a room with multiple saibamens and having to kill them all. The saibamen were weak compared to Vegeta so fighting enemies weaker than him didn't help increase his strength and he didn't develop new techniques when fighting them. This point can also be used to describe his father, King Vegeta. As the King he already climbed to the top by having to defeat enemies and even other saiyans that got in his way but when he made it to the top Vegeta didn't have any ways to train so he was at a standstill and never able to reach his true potential.
Now as a saiyan their only way to get stronger was using the Zenkai boost i.e. being in a near death experience. But think about how dangerous this was, when you're only way to get stronger is to almost die in battle you weren't going to survive many battles which is why most saiyans wouldn't have made it to very old age with the exception of King Vegeta until his death so because of this the ability to fight for many years without your body growing old (extreme durability) wasn't useful because most of them would have die from the near death ability they had to achieve in battle to get stronger.
Also look at the saiyans ranking system, they believed if you were weak at birth you were going to be weak and there was no way around that. I can argue that saiyan culture was most like similar to Spartan where if you couldn't be shipped to another planet as a weak baby they would've probably executed you at that young age which limited their chances of getting someone ingenuitive who could help develop the training techniques to get stronger without having to have a near death experience. They were sent to planets and were set loose without any training or preparation they weren't even given clothes or resources to survive. The whole ideology is "this is a saiyan baby and if set loose on the planet with the goal of demolishing it he would grow stronger than the beings on this planet". And if the saiyan baby couldn't demolish it their own natural ability the Oozaru state would do the job for them/ the process of the full moon would take care of itself. But due to having no training whatsoever they couldn't control the giant ape state so they were just rampaging across the planet.
The scouters were the worse thing that could've ever happen to the saiyan race. The scouters were useful for tracking down enemies across the planet but if you look at it from another standpoint the scouters just became a way to ranks saiyans to 1st class, 2nd class, or 3rd class warriors. So the saiyan race began to believe that your power level at birth determined you potential and told how strong you were going to be. This is an ideology we frown upon as humans because it's basically saying your "power is decided the day you are born". They saw your power level and they ranked it which stereotyped you the day you were born.
Also if you look at it in the view point of Broly the power level created a stereotype that you were going to be a monster if your power level was too high. Now I know that in the movie Broly was insane and crazy but imagine if he was raised on Earth there's a chance that this crazy urge he had could be controlled like Goku's was which would give our planet a warrior that was insanely strong and could help improve and protect our planet.
So the scouters' created the stereotype that you were either weak-average-elite and there was no concept of rising to the top with hard work and dedication. Now going back to the Spartan ideology this power level concept was used to snuff out weak babies and just like in Spartan concept low power level was most likely seen as a defect and you were killed.
So the question of why the saiyans weren't the strongest in the universe when they had states like Super Saiyan 3 and Super Saiyan God is rooted to their culture. In Goku and later Vegeta it was evidence that saiyans had the biology and the potential but their culture didn't teach the concept of training and becoming stronger through hard work and developing new techniques for combat.
The point is unless they were exposed to outside culture (like Goku and later Vegeta) they would've just been a very barbaric race of mercenaries like the Spartans mildly stronger barbarians but they wouldn't have gotten anywhere.
Evidence of this is the multiverse saga where they show a race of saiyans without Freiza and they were embarrassingly weak when compared with a Vegeta who was exposed to Earth's culture. Also if you look at Vegeta's life he experienced the strongest power boost of his life when he was exposed to Earth's culture further proving even if at an older age if saiyans were given Earth's culture they would be a very, very powerful race.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Just a one shot of a theory I got from Stephen Ross at "real-anime-training" you can watch his video online because he broke it down better and really explains step by step the whole theory.