Does this prove that the children of converts are not Israelites?

Question

I was under the impression that even though a woman who had a real(halachic) conversion was not an Israelite, her children would be full fledged Israelites in every respect, indistinguishable in matters of Torah Law from those Israelites who are not descended from converts.

But the Mishna in Horayos 3:8 says that an Israelite takes precedence over a Natin. (A Natin is a descendent of the Gibeonites who converted in previous generations).

Now I know that as we don't know who is a Natin in this generation, any Jew who is not a Kohen or a Levi is an Israelite.

But does the above mentioned Mishna prove that the children of converts are not Israelites?

Source: https://www.sefaria.org/English_Explanation_of_Mishnah_Horayot.3.8.2?lang=bi&with=all&lang2=bi

Answer 1

The Tiferes Yisroel Yachin 35 there answers your question. Long story short, in some ways the Nesinim are worse than regular converts. As such, the descendants of converts are full fledged Jews, like you said, unlike the descendants of the Nesinim.

וממזר לנתין הן הגבעונים שהטעו את יהושע. שנקראו נתינים, ע"ש שנתנן יהושע לחוטבי עצים ושואבי מים. וזרעם פסול לקהל, לרמב"ם [פי"ב מא"ב] מדרבנן, ולרמב"ן ורשב"א ותוס' ריש אלו נערות, אפילו כשנתגיירו אסורים בחיתון מדאורייתא, ורק בנים שנולדו להן אחר שנתגיירו, אסורים מדרבנן...‏

A mamzer takes precedence over a Nasin. They're the Gibeonites who tricked Yehoshua (into forming a covenant). They're called Nesinim because Yehoshua decreed that they'd be woodchoppers and water drawers. Their descendants are disqualified to marry into the Jewish people. According to the Rambam this is Rabbinic. According to the Ramban, Rashba and Tosafos even if the Nesinim converted properly it is Biblically forbidden for them to marry into the Jewish people. However their descendants after they converted are Rabbinically forbidden...

So we can't learn anything from them to regular converts.