The 1917 Code's canon 984 says

Illegitimate ones, whether the illegitimacy is public or occult, unless they were legitimated or professed solemn vows

are "irregular by defect".

Why doesn't the 1983 Code Can. 1041 prevent illegitimate children from entering the clerical state or continuing its exercise?

The 1917 Code's canon 984 says" Illegitimate ones, whether the illegitimacy is public or occult, unless they were legitimated or professed solemn vows"

Why doesn't the 1983 Code Can. 1041 prevent illegitimate children from entering the clerical state or continuing its exercise?

The 1917 Code of Canon law did allow illegitimate children to enter the seminary and become priests if these men could obtain a dispensation.

It even allowed postulants to enter the novitiate and become professed nuns by dispensation.

When I was a postulant petitioning to enter the novitiate in the mid 1960s, the nuns asked me if there was anything irregular that would prevent me from entering the novitiate and taking my vows according to the 1917 Code of Canon Law. There was a list of five items, but I was in the clear. We had to study those codes. In our discussions preparing to enter the novitiate, the novice mistress told us that several postulants had to get dispensations from the bishop or from Rome, but that the process had started long before they entered the convent as postulants, so they were good to go.

I talked with several men who received dispensations. One man told me that he had been adopted by the Pope so that he could enter the seminary. Such papal adoptions clear up children who are illegitimate. And back in 1960, those under 21 were considered to be children, so the pope could adopt them. Many seminarians entered as high school freshmen, so they were only 13 or 14 years old. Entering the college seminary, they would be between 16 to 19 years of age, still considered to be children.

With the apocalyptic shortage of good priests at the moment, why on earth would you ask such a question?
I personally know a fantastic Traditional Catholic family who started out in the Novus Ordo, but later came to Tradition. They have always homeschooled their children and their oldest son, a very devout young man (doesn’t even own a cell phone at 18 )  would be perfect for the seminary. However, as you stated he should not be considered because unfortunately he was born before his parents were married.

How humiliating it would be for him to be turned away from a seminary not knowing that one.

Sad I didn’t even know.

He can get a dispensation. These dispensations are granted if the candidate for the priesthood or religious life is known for living a devout life.

The parents would have to discuss this with their bishop.

he should not be considered because unfortunately he was born before his parents were married.

To become a priest, he can be legitimated if his parents legitimate their marriage, or he can first become a religious and takes solemn vows.

It was a mere formality in the 1917 Code to dispense this defect from being an impediment to ordination.  So in practice it makes little difference.  But in PRINCIPLE it upheld the notion that there is indeed something DEFECTIVE about being born out of wedlock.

Born illegitimately does not equal poorly raised.

Being illegitimate can be a handicap (just like bodily irregularities (missing thumb, epilepsy, severe speech impediment, mutilation, etc.) prevent some men from becoming priests, unless a dispensation is sought).

More than half the Traditional Catholics (and some Novus Ordo) that I know are converts to the Faith. Even HE +Will is a convert.

I'm confused how this is related to illegitimacy.

This.

Also, it's not a small matter to be without a father. We learn about God and authority itself from our fathers. If our fathers are absent or gravely defective, our notion of God ("Our Father in Heaven") becomes warped.

The modern world acts like fathers are optional -- take em or leave em. They couldn't be more wrong.

And as we see in statistics for fatherless children, children without a father figure are several times more likely to turn to drop out, turn to crime, become drug addicts, and father illegitimate children themselves. It is statistics like that which remind anyone of sense that the laws God gives us are not there just for His benefit, but also for our own. Marriage isn't just some arbitrary institution as liberals would have us believe, it's a union that unites the wedded and serves as a glue for the family, ensuring the long-term stability and happiness of all the family. And that in turn leads to the children being raised well and disciplined. Modernists act as if marriage is a limitation and that people are so much happier and "liberated" when let fornicate, but every statistic shows that men and women with multiple life-time sɛҳuąƖ partners are far less happy than those with just 1, with their unhappiness increasing with the number of partners. And as we know from fatherlessness statistics, it harms their children too.

The world has simply forgotten discipline. They think that because their mindless hedonism feels good in the short-term that it is good. But fornication, onanism, etc. can be compared to food in a way. Yes, gorging on McDonalds may feel very nice, but as everyone knows it's very unhealthy. And all the crimes against natural law are just the same, while they may feel good to fallen man in the short-term, in the long-term they'll only give you misery. If only those unmarried mothers realised this.

Bergoglio seems to have had a fatherless upbringing:
From The Dictator Pope:

Jorge Mario Bergoglio was born on December 17, 1936, in a suburb of Buenos Aires, the son of a struggling accountant. The signs of strain that can be detected in his family are not merely economic. The adult Jorge was not given to speaking of his parents. After the birth of her fifth child, his mother became temporarily an invalid and had to delegate the upbringing of her children to a woman called Concepción. Jorge celebrated this surrogate as a good woman, yet he admitted that he treated her badly when, years later, she came to him to ask for his help as bishop in Buenos Aires and he sent her away, in his own words, “quickly and in a very bad way.”¹ The incident seems to point to strains which are buried in the past but may provide a clue to Bergoglio’s enigmatic personality.