Is a child born out of wedlock still considered as an illegitimate child if his / her biological parents got married?

Committing adultery or premarital sex is a great act of sin in Islam. The person has to repent and seek forgiveness from God.
The child born out of wedlock is regarded as an illegitimate child, although if the biological mother eventually marries the biological father.
As according to a hadith narrated
"الولد للفراش"
Which means: "The child belongs to the bed (marriage relationship)" (Narrated by Imam Abu Huraira)
As the child is attributed to the mother and not the father, hence, the father is not eligible to stand in as a wali for the illegitimate child during marriage. And according to Islamic ruling, it is not allowed for the child to carry the father’s name, and they do not inherit from each other.