# Why are a plow's metal rings used to connect oxen considered integral?

The 21st chapter of tractate Keilim discusses which attachments/pieces of utensils are considered integral so as to be considered connected for tumah purposes. My understanding of the underlying principle of this chapter is that if we reduce the object to its minimum requirements, those attachments that are still required for its function are considered connected, and those that are not, are not.

For example, in mishnah 3, we see that a hand-saw requires its handles, but does not require its stabilisation pieces (even though it would be harder to use without them). In mishnah 1, we see that the material that is essential to the shirt (and will become part of the shirt) is attached, but the loom itself is not attached (since clothes can be sewn without a loom, even though it's harder).

Mishnah 2 discusses a plow:

הַנּוֹגֵעַ בָּעֹל, וּבַקַּטְרָב, בָּעַיִן, וּבָעֲבוֹת, אֲפִלּוּ בִשְׁעַת מְלָאכָה, טָהוֹר. בַּחֶרֶב, וּבַבֹּרֶךְ, וּבַיָּצוּל, טָמֵא. בָּעַיִן שֶׁל מַתֶּכֶת, בַּלְּחָיַיִן, וּבָעֲרָיִין, טָמֵא. רַבִּי יְהוּדָה מְטַהֵר בַּלְּחָיַיִם, שֶׁאֵינָם עֲשׂוּיִן אֶלָּא לְרַבּוֹת אֶת הֶעָפָר:

One who touches the yoke, the crossbar, the collar-piece, or the thick ropes, even when they are being used, he remains clean. If he touched the tail piece, knee or handle, he becomes unclean. If he touched the metal rings, the guides, or the flanks, he becomes unclean. Rabbi Judah says that he remains clean if he touched the guides since they only serve to increase the soil. [from Sefaria]

The "tail piece", "knee", "handle", "guides", and "flanks" are essential pieces of the plow that make it do its job (except to R. Yehudah, the "guides" perform a function that is non-essential). The "metal rings" are only there to connect the oxen to the plow. However, plows can (and have been) pulled without oxen, so this would seem to be a non-essential part of the plow, when reduced to its minimum requirements.

According to how I've described this, why should the "metal rings" convey tumah? Is my understanding of the underlying principle flawed?