divorce - What problems are faced by children born out of wedlock?

I was previously under the impression that, since a man may acquire a woman with sexual intercourse (cf: Mishna, Qiddushin 1:1), if a couple have a son but no ketubah, the child still inherits his father's property. See, for example, the mishna in Ketubot 4:9, which stipulates that sons inherit whether that detail is included in the ketubah or not. Recently, however, I learnt a mishna that appears to imply the opposite: that sons born out of wedlock face some manner of "damage", even though the identity of their father is known.

My question concerns whether or not I am learning this material properly: does the term that I have translated as "damage" (פגם) refer to their inability to collect on their father's inheritance? If not, might it be referring to the notion that it is somehow socially damaging to suggest that somebody was born outside wedlock? Why would it be an insult if their parents were living together at the time of their conception?

The mishna in question, together with the relevant peirushim, is as follows:

בית שמאי אומרים: פוטר אדם את אשתו בגט ישן ובית הלל אוסרין. איזהו גט ישן? כל שנתיחד עמה אחר שכתבו לה

The school of Shammai says that a man may divorce his wife with an old document, but the school of Hillel forbids it. What is an old document? Any [situation] in which he was secluded with her after writing it.

• Gittin 8:4

As R' Menachem Meiri points out, the mishna is speaking of a document (a get, as it is colloquially known) that was both written and signed on the day in question, but which was not given to her until a period of time had elapsed. The reasoning for prohibiting such a document is explained by Rav Ovadiah of Bertinoro (based on the gemara in Gittin 79b) as follows:

גזירה שמא יאמרו גטה קודם לבנה שמא ישהה את הגט שנה או שנתים בין כתיבה לנתינה ויהיו לה בנים ממנו בתוך זמן ואח"כ יגרשנה בו וכשיראו זמן הגט קודם ללידת הבן יהיו סבורים שנתן לה גט משעת כתיבה והוי פגם שיאמרו מן הפנויה נולד

[This is an] enactment, lest people suspect that the document precedes [the birth of] her son. Should there be a period of one or two years between the writing [and signing - Meiri] of the document and its being given, and should she have children to him within that time, and should he divorce her with that document afterwards, then when people see that the document predates the birth of her son they will think that he gave her the document at the time of its composition, and there will be damage - for they will say that he [her son] was born of an unmarried woman.

• Bartinura, ibid.

(The full quote from Rav Ovadia includes the words "ב"ש סברי לא אמרינן" at the beginning, noting that the school of Shammai permit an old document because they don't hold by this enactment.)