Our Gemara on Amud Aleph tells us that according to Rabbi Yehuda, the congregation of converts is called the congregation, and therefore subject to similar marriage restrictions as non-converts.
What is a “congregation” (קהל)?
The Shalah (Torah Shebikhsav, Chukas, Torah Ohr) explains that there are four kinds of “congregations” within the Jewish nation as used in scripture:
Am (עם), Kehal (קהל), Eidah (עדה), and Khal Hashem (קהל ה׳).
The “Am” refers to a basic level of mentality and consciousness. This class of people are the same as the Mixed Multitude that joined up with the Jewish people during The Exodus from Egypt.
Kehal, is as described in our Gemara. Fully committed Jews, but composed of converts.
Eidah is composed of full Jews, born as Jews.
Kehal Hashem is the highest level of Jews who possess a soul that is capable of reuniting with its original state from which it was disengaged at its inception. (I don’t really understand what the Shalah means by this last point, but I did my best to translate it.)
The Shalah uses these four categories to explain the sin of the waters of rebellion described in Bamidbar (20). In this section of the Torah, the people are alternately described with different words for congregation.
וַיָּבֹ֣אוּ בְנֵֽי־יִ֠שְׂרָאֵ֠ל כׇּל־הָ֨עֵדָ֤ה מִדְבַּר־צִן֙ בַּחֹ֣דֶשׁ הָֽרִאשׁ֔וֹן וַיֵּ֥שֶׁב הָעָ֖ם בְּקָדֵ֑שׁ וַתָּ֤מׇת שָׁם֙ מִרְיָ֔ם וַתִּקָּבֵ֖ר שָֽׁם׃
The Israelites arrived everybody at the wilderness of Zin on the first new moon, and the people stayed at Kadesh. Miriam died there and was buried there.
וְלֹא־הָ֥יָה מַ֖יִם לָעֵדָ֑ה וַיִּקָּ֣הֲל֔וּ עַל־מֹשֶׁ֖ה וְעַֽל־אַהֲרֹֽן׃
The community was without water, and they joined against Moses and Aaron.
וַיָּ֥רֶב הָעָ֖ם עִם־מֹשֶׁ֑ה
The people quarreled with Moses
וְלָמָ֤ה הֲבֵאתֶם֙ אֶת־קְהַ֣ל ה׳ אֶל־הַמִּדְבָּ֖ר הַזֶּ֑ה לָמ֣וּת שָׁ֔ם אֲנַ֖חְנוּ וּבְעִירֵֽנוּ׃
Why have you brought ה׳’s congregation into this wilderness for us and our beasts to die there?
וַיָּבֹא֩ מֹשֶׁ֨ה וְאַהֲרֹ֜ן מִפְּנֵ֣י הַקָּהָ֗ל אֶל־פֶּ֙תַח֙ אֹ֣הֶל מוֹעֵ֔ד וַֽיִּפְּל֖וּ עַל־פְּנֵיהֶ֑ם
Moses and Aaron retreated from the congregation to the entrance of the Tent of Meeting, and fell on their faces.
וַיְדַבֵּ֥ר הָ֥ אֶל־מֹשֶׁ֥ה לֵּאמֹֽר
And Hashem spoke to Moses, saying,
קַ֣ח אֶת־הַמַּטֶּ֗ה וְהַקְהֵ֤ל אֶת־הָעֵדָה֙ אַתָּה֙ וְאַהֲרֹ֣ן אָחִ֔יךָ וְדִבַּרְתֶּ֧ם אֶל־הַסֶּ֛לַע לְעֵינֵיהֶ֖ם וְנָתַ֣ן מֵימָ֑יו
You and your brother Aaron take the rod and assemble the community, and before their very eyes order the rock to yield its water.
וַיַּקְהִ֜לוּ מֹשֶׁ֧ה וְאַהֲרֹ֛ן אֶת־הַקָּהָ֖ל אֶל־פְּנֵ֣י הַסָּ֑לַע
Moses and Aaron assembled the congregation in front of the rock;
יַ֚עַן לֹא־הֶאֱמַנְתֶּ֣ם בִּ֔י לְהַ֨קְדִּישֵׁ֔נִי לְעֵינֵ֖י בְּנֵ֣י יִשְׂרָאֵ֑ל לָכֵ֗ן לֹ֤א תָבִ֙יאוּ֙ אֶת־הַקָּהָ֣ל הַזֶּ֔ה אֶל־הָאָ֖רֶץ אֲשֶׁר־נָתַ֥תִּי לָהֶֽם׃
Because you did not affirm My sanctity in the sight of the Israelite people, therefore you shall not lead this congregation into the land that I have given them.”
Commentaries throughout the generations have tried to pinpoint what the exact sin was that caused Moshe and Aharon to be denied entry into Eretz Yisrael, as the scriptural narrative is ambiguous. Some say it was because Moshe hit the rock instead of speaking to it, thereby decreasing the miracle (see Rashi Op. Cit 20:12) or because Moshe became angry and called the Jews Rebels (See Shemoneh Perakim 4). The Shalah offers his own unique explanation. Only the part of Israel known as the Kehal Hashem are able to experience the full miraculous powers of the YKVK that upends nature. When you have Jews that include the Khal Hashem and the Eidah, they too may witness miracles and are called collectively Bnai Yisrael, since quite literally, they are the two groups who are descendents of Yaakov, unlike the Eirev Rav and the Converts.
The Am complained to Moshe and Aharon, while their argument was more subtle. The Am said, “Perhaps we, the Am, do not deserve miracles, but how could you let the Khal Hashem starve in the wilderness!”
Then Hashem tells Moshe and Aharon, “Gather the Eidah so they may witness the miracle of the water.” Hashem is allowing even the Eidah to witness this great miracle, but not the Khal nor the Am. Moshe and Aharon err, and include the Khal as witnesses. According to the Shalah, this was this sin of Moshe and Aharon, that they included the Khal as witnesses for this miracle, when they were undeserving. God then tells them:
יַ֚עַן לֹא־הֶאֱמַנְתֶּ֣ם בִּ֔י לְהַ֨קְדִּישֵׁ֔נִי לְעֵינֵ֖י בְּנֵ֣י יִשְׂרָאֵ֑ל לָכֵ֗ן לֹ֤א תָבִ֙יאוּ֙ אֶת־הַקָּהָ֣ל הַזֶּ֔ה אֶל־הָאָ֖רֶץ אֲשֶׁר־נָתַ֥תִּי לָהֶֽם׃
Because you did not affirm My sanctity in the sight of the Bnai Yisrael, therefore you shall not lead this congregation (Khal) into the land that I have given them.”
Since Moshe and Aharon did not restrict display of the miracle, and allowed the Khal to see it as well, instead of only the Bnai Yisrael (Eidah and Khal Hashem), they would not be able to accompany the Khal in entering the Promised Land.
It comes out that according to the Shalah, the sin was not hitting the rock, and not letting the Jews see enough of a miracle, but rather for allowing too many unauthorized people to see the miracle.
This might explain the what, but not the why. What is the objection for the non-Jews to have seen this miracle? What is so terrible? Though the Shalah does not explain this explicitly, we can deduce it by keeping track of who was complaining and rebelling. The Am was complaining, not the Khal Hashem nor the Eidah. So the complainers should not merit to see the miracle as they should have had more faith. The only difficulty with this, is that the Khal was also excluded and they were not the complainers. I’ll add, Moshe and Aharon probably thought to include the khal but exclude the Am, precisely because of this point, that the Am complained but not the Khal. However, upon closer examination, we see that Moshe and Aharon were also intimidated by the Khal. They do not say anything, but from the context of the verse, we can see that they are acting upon the backing and complaints of the Am, therefore they are to be blamed as well. As the verse reports:
וַיָּבֹא֩ מֹשֶׁ֨ה וְאַהֲרֹ֜ן מִפְּנֵ֣י הַקָּהָ֗ל אֶל־פֶּ֙תַח֙ אֹ֣הֶל מוֹעֵ֔ד וַֽיִּפְּל֖וּ עַל־פְּנֵיהֶ֑ם
Moses and Aaron retreated from before the congregation to the entrance of the Tent of Meeting, and fell on their faces.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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