Short answer - Shastras do not recommend Kshatriya-woman marrying Suta-man. So she rejected him.
Loong answer (with points & counter-points):
According to public knowledge, Karna was a Suta-putra.. except Kunti none else knew his real birth as son of Surya deva with Kunti solely through use of Durvasa mantra (without any physical union).
Suta means charioteer. Charioteer's post is in between fighting (kshatriya's duty) and advising (brahamana's duty) - He must be strong enough to withstand arrows. He must also know the battlefield well enough to navigate it, his master's strengths/weakness as well as enemy's strengths/weakness in order to provide advice to both defend and attack. He need not be strong enough to hurl arrows, nor he have enough knowledge to actually teach his master how to shoot them.
So, the best qualified man for this job should have both kshatriya blood and brahmana blood mixed - Yes, DNA does imbibe traits in birth. Scientifically proven, at least in animals (Race-winning horses and fight-winning bulls are set out to stud/mate). Which means his father and mother must be of those 2 different varnas.
Now, going by common sense, one may argue that it can either be Brahmana-father + Kshatriya-mother, or other way Kshatriya-father + Brahamana-mother. But, that is not the case. A Brahmana's son usually won't have the necessary strength/valor to face arrows in battlefield, and his knowledge/penance are better suited to serve the country in other areas.
Now, going by shastras, the first type (Brahmana-father + Kshatriya-mother) is anuloma (recommended) according to shastra. Second type (Kshatriya-father + Brahamana-mother) is Pratiloma (not-recommended) according to shastra. The son born from this particular pratiloma combination is called Suta.
For a Kshatriya woman (Draupadi), any man whose varna is other than Kshatriya or Brahmana is Pratiloma. Since Karna was the son of a Suta, society thought and expected him to belong to same Suta varna. So she rejected him.
The different combinations of varna-sankara (mixing) and their progeny are given here : https://iskcondesiretree.com/profiles/blogs/types-of-marriages
(Note - this is not direct scripture, but it points to Manu Smriti. If anyone has direct link, feel free to edit)
Common folk might cry - 'But VALOR ALONE is important' - first of all, right after this swayamvara, everyone tried to fight Arjuna and lost, including Karna. He also lost to Arjuna during Virata battle. So, compared to Arjuna, he is not better. Secondly, if only valor was important, Ravana had super-valor, but do we praise him ? No, we burn his effigy even now couple days back at Ram Lila. So valor alone is not important.
Now the common folk will cry again 'These 2 points only show that Karna was not valorous enough and that Ravana's character was bad. But during swayamvara, Draupadi neither knew about Karna's valor nor his character, meaning she rejected him solely based on caste, without even giving him a chance'
These same common folk, when afflicted with a strange disease, will accept a medicine when given by a doctor, but will reject the same medicine if the man was a bank-clerk or police-constable. Ask them why they do so? Whatever reason they give, give the same reason back to them.
They will cry back 'Taking medicine is not same as Shooting rotating fish. Medicine from untrusted man involves risk to me. Whereas Karna shooting a random fish does not risk Draupadi. So she can safely check his valor without risk. Then she can check his character (Ravana's example). If both are good, then she can marry.'.
This is like saying 'Have lab-animals test the medicine without risk to you. Then check if there are any side-effects/allergies. If both are good, then take medicine'
Alright. How would you go about checking someone's character, or checking medicine's side-effects ? Live with him for a while like modern-day girlfriend. This is same as taking a little bit of the medicine. Both are out of question because of risk involved.
So we are left with - asking other people who we trust about the person and about the medicine. Now we reach the crux of the matter.
She did ask elders and trusted people and knowledge books and research papers and experts in the field (aka RISHIS). They all said, THERE WILL MOST LIKELY BE SIDE-EFFECTS. Marrying a SUTA-man is not recommended for KSHATRIYA-woman. Valor alone is NOT ENOUGH. Even if the valor existed for the period of the rotating-fish test, we don't know if it will last a lifetime. Same could be said for Arjuna, but character and Compatibility and Force of dynasty/blood-line are also important. Even if a person wanted to do a bad thing, the fear of bringing shame to the name of his long-chain of famous forefathers will prevent him doing it. This shame may not be there for a man who didn't have illustrious ancestry.
Why, even today, they set race-winning horses or bulls to stud, so that the DNA can propagate. Why not let a random horse or bull mate, and then check whether the calf can run/fight - depending ONLY on valor, not on birth? Because they don't wan't to risk it. And even if the horse/bull could fight, we don't know how long it might last.
Because Birth is tied very closely with valor, character, demeanor, compatibility etc. - all points that even a modern day bride looks for in a man. If a person has just one aspect, but you can't be sure of all, then you likely won't risk it.
Valor alone is not enough. Valor displayed in a tournament alone is not enough. Arjuna already had the victory over Drupada under his belt, when he was just a student. Karna had no such victories. Draupadi didn't know if Arjuna was alive, nor if he would even show up to swayamvara. Still, according to elders, Karna was a known risk, because he was Suta-putra and it was Pratiloma. Hence she would rather wait for an unknown variable than accept a known risk. Even if some other Suta, or Vaishya had taken up the bow, she would have still said no. Even if it was some other Kshatriya, with proven valor, she could have still said no based on her personal preference. Some might say 'she only gave the excuse of Suta-putra, when in fact, she didn't like him. If some other varna man proved his valor, she might have accepted him' - Both reasons are perfectly acceptable to shastra - whether rejecting based on varna or personal preference. Her personal preference was Arjuna (she was literally born from sacrificial fire directly at age of 16 to marry Arjuna). But the man who won the tournament was a Brahmana, which is Anuloma, hence she married even without bothering if it was Arjuna or not, later only they all found out the Brahmana was actually Arjuna.