I'm a little nervous about writing an answer as it's my first time, but I'll give it a go!

To answer this you have to look at both Dudley as an individual and the larger picture of both English and European Politics at the time.

Marrying a courtier would have cause issues amongst the court. By marrying Dudley, she would be picking a specific family within the court to elevate and favour, meaning that his friends and family would have had greater influence within court compared to other noble families.

This meant that the most sensible thing to do would be to marry a foreigner, such as her other suitors like Archduke Charles of Austria, or the French Duke of Alencon. There was also political benefit in marrying into a foreign alliance in order to improve England's influence upon the European playing field. As a result, an internal marriage was less encouraged.

Furthermore, Dudley and Elizabeth's overt affection for one another caused issues in the negotiation of these potential marriages as it seemed clear to foreign ambassadors that the Queen had already chosen a consort.

Although Dudley had redeemed himself fighting in France and gained prestige, his father John Dudley had unsuccessfully lead the Lady Jane Grey conspiracy to prevent Mary I becoming Queen, and as such their family name was tarnished.

Moreover, Dudley's first wife Amy Dudley befell an untimely death having fallen down the stairs and broken her neck, and paired with Elizabeth's refusal to marry and the rumoured dalliances between Elizabeth and Dudley, there was much rumour of foul play.

In conclusion, for some their dislike of Dudley was down to personal grievances in relation family status and dirty rumour, but also the risk of losing out on a European marriage.

I can provide sources if requested, I hope this helped :)

Edit to add: Most European powers were also Catholic at the time. A marriage to a Catholic leader would have lessened the chance of England being targeted as a Protestant country, whereas a marriage to another Protestant noble would have solidified their status as an outlier like the Dutch.

The Elizabethan court would have seen Dudley as an entirely unsuitable match on essentially every possible level.

Robert Dudley's grandfather, father, and brother were all attained as traitors and duly executed. Granted, by the Tudor age it may have been hard to find a family tree without some traitors in the branches. Certainly not Elizabeth's own family on the mother's side. But the Dudleys were a relatively new family, which was another mark against them. And his father was truly despised for his attempted coup with Jane Grey.

A family of traitors, and a new family at that, is not the ideal hunting ground for a consort for a queen-regnant.

At time, marriage of any great personage, never mind a Queen, was not a love match. It was supposed to consolidate power and bring advantages to both parties. Dudley had nothing to offer Elizabeth. His family had lost most of their wealth and prestige under Mary and rightly so. Elizabeth rewarded both Robert and his brother Ambrose (and to a lesser extent his sister Mary Sidney) with titles, land, monopolies, positions of power. Everything they had they received from the Queen. What did they have to offer in return?

Elizabeth's father was an oddity in that he did want a love match but there is nothing to suggest Elizabeth genuinely believed this was a possibility for herself.

Most of her council preferred a foreign match to bring an alliance which would protect the weak and financially destitute kingdom. Although they differed strongly about who that should be. Elizabeth, herself, only ever seriously pursued foreign matches.

Dudley was also already married. Which presents a rather obvious hurdle to the whole thing.

When his wife did die it was under what was considered suspicious circumstances (she was found with her neck broken at the bottom of the stairs.) Many thought Dudley killed her to make room for the queen. Some thought she killed herself out of grief over their affair, which is in some ways worse. It was a scandal through out Europe. Elizabeth acted quickly and had him sent away from her. And demanded a full and thorough investigation. Although he was exonerated there was not any question that she could marry him now.

There was also the concern of factionalism. If Elizabeth married an Englishman, any Englishman, then he would likely set up his own faction in court favoring his own family and retainers. While this would be true of any Englishman, Dudley was particularly prone to setting up his own faction already. The Dudley faction and the Cecil faction were often at odds though out Elizabeth's reign. This would only be exacerbated if Elizabeth married him. The Cecil faction would never be comfortable with it. And Elizabeth did listen to William Cecil.

Also, frankly, many people at court just plain didn't like him. Others did like him. He was charming, witty, and intelligent in his own way. But many others saw him as arrogant, grasping, and greedy.

From the point of view of a gentleman in Elizabeth's court, Robert Dudley had nothing to offer. Perhaps he would have made Elizabeth happy, although I am skeptical, but that couldn't be lower on a courtier's list of priorities when it came to marriage.

Whether they had an illegitimate son together? This can be pretty adamently dismissed. As Elizabeth, herself, remarked she was never alone. She was constantly surrounded by her ladies. She didn't even sleep alone instead sharing the bed with one or two of her most trusted ladies (which was common at the time.) Once queen, in addition to the above, she was also daily with the men of her court. Her dresses were unforgiving and she rather famously maintained a slim figure her entire life. If she had ever had a child it would be known. Her pregnancy would have been obvious. People in Tudor times gossiped to an extent that would make high school kids blush.

The illness Elizabeth had in 1561, as referenced in that Wikipedia article, was well documented. It was small pox. She was well attended during this period by doctors and her ladies. It left marks on her body, although not her face as often claimed. She was also only bed ridden for a few weeks. She would still have had to hide a pregnancy for eight months before that, which would have been impossible for the reasons stated. It should be noted that even the Spanish, who by the 1580s were Elizabeth's bitterest enemies, did not believe Arthur Dudley's story.