Is there any connection between the Sotah, the Mizbeiach, and the Beis HaMikdash keychain?
In exactly three contexts, the Mishnah notes that there was an amah square marble slab with a ring, using exactly the same language each time.
The first is in Sotah 2:2, regarding the Sotah procedure:
נִכְנַס לַהֵיכָל וּפָנָה לִימִינוֹ, וּמָקוֹם הָיָה שָׁם אַמָּה עַל אַמָּה, וְטַבְלָא שֶׁל שַׁיִשׁ, וְטַבַּעַת הָיְתָה קְבוּעָה בָהּ. וּכְשֶׁהוּא מַגְבִּיהָהּ, נוֹטֵל עָפָר מִתַּחְתֶּיהָ וְנוֹתֵן כְּדֵי שֶׁיֵּרָאֶה עַל הַמַּיִם
He entered the Heichal and turned to his right. There was a place there, one amah by one amah, with a slab of marble, and a ring was fixed on it. When he would lift it, he would take dirt from underneath it and place enough that it can be seen on the water.
The second is in Middos 1:9, regarding the procedure to lock up the Beis HaMikdash; the previous Mishnah identifies this place as the Fire Chamber:
וּמָקוֹם הָיָה שָׁם, אַמָּה עַל אַמָּה, וְטַבְלָא שֶׁל שַׁיִשׁ וְטַבַּעַת הָיְתָה קְבוּעָה בָהּ, וְשַׁלְשֶׁלֶת שֶׁהַמַּפְתְּחוֹת הָיוּ תְלוּיוֹת בָּהּ. הִגִּיעַ זְמַן הַנְּעִילָה, הִגְבִּיהַּ אֶת הַטַּבְלָא בַּטַּבַּעַת וְנָטַל אֶת הַמַּפְתְּחוֹת מִן הַשַּׁלְשֶׁלֶת, וְנָעַל הַכֹּהֵן מִבִּפְנִים
There was a place there, one amah by one amah, with a slab of marble, and a ring was fixed on it, and a chain on which the keys would hang. When the closing time came, he would lift the slab by the ring and take the keys from the chain, and the Kohen would lock from the inside.
The third is in Middos 3:3, regarding the pit under the Mizbeiach where the bloods would flow into the Kidron:
לְמַטָּה בָרִצְפָה בְּאוֹתָהּ הַקֶּרֶן, מָקוֹם הָיָה שָׁם אַמָּה עַל אַמָּה, וְטַבְלָא שֶׁל שַׁיִשׁ, וְטַבַּעַת הָיְתָה קְבוּעָה בָהּ, שֶׁבּוֹ יוֹרְדִין לַשִּׁית וּמְנַקִּין אוֹתוֹ.
Under the floor in that corner, there was a place there, one amah by one amah, with a slab of marble, and a ring was fixed on it, through which they would descend into the pit and clean it.
Is it just that these three things require access to underneath, and this was an elegant way to provide access without looking ugly? Or is there something deeper (so to speak) which connects these three things?