We learn in the Talmud (Pesachim 66a)
אמרו לו "רבי, שכח ולא הביא סכין מערב שבת מהו?"
אמר להן "הלכה זו שמעתי ושכחתי אלא הנח להן לישראל אם אין נביאים הן בני נביאים הן"
They said to Hillel: Our teacher, if one forgot and did not bring a knife on the eve of Shabbat and cannot slaughter his Paschal lamb, what is the law?
He said to them: I once heard this Halacha from my teachers but I have forgotten it. But leave it to the Jewish people; if they are not prophets to whom God has revealed His secrets, they are the sons of prophets, (and will certainly do the right thing on their own).
למחר מי שפסחו טלה תוחבו בצמרו מי שפסחו גדי תוחבו בין קרניו ראה מעשה ונזכר הלכה ואמר "כך מקובלני מפי שמעיה ואבטליון"
The next day, (on Shabbat that was the eve of Passover), one whose Paschal offering was a lamb took the knife and stuck it in its wool; and one whose Paschal offering was a goat, which does not have wool, stuck it between its horns. Hillel saw the incident and remembered the Halacha that he had once learned and said: "This is the tradition I received from the mouths of Shemaya and Avtalyon", (meaning that this is, in fact, the proper course of action).
And this is codified by the Rambam (Hil. Korban Pesach 1:19)
שָׁכַח וְלֹא הֵבִיא סַכִּין לֹא יְבִיאֶנָּה בְּשַׁבָּת. אֶלָּא נוֹתְנָהּ בֵּין קַרְנֵי הַכֶּבֶשׂ אוֹ בְּצַמְרוֹ וּמַכִּישׁוֹ עַד שֶׁמְּבִיאוֹ לָעֲזָרָה וּמַקְדִּישׁוֹ שָׁם. וְאַף עַל פִּי שֶׁהוּא מְחַמֵּר בְּשַׁבָּת מְחַמֵּר כִּלְאַחַר יָד הוּא וּמִפְּנֵי הַמִּצְוָה מֻתָּר.
If [Erev Pesach is on Shabbat and] a person forgets to bring his knife [to the Beit Hamikdash before Shabbat], he may not bring it on Shabbat.
Instead, he should entangle it in the animal's horns or fleece and have the animal bring it for him. This is permitted in this case since it's needed for a Mitzvah.
Why would a person need to bring a knife to the Beit Hamikdash? Didn't they have a supply of them? They slaughtered animals all the time.
Furthermore, we don't see anywhere else that you were expected to bring your own knife.
Why was one allowed, nay, expected, to bring one's own knife, even on Shabbat?
To further clarify and sharpen the question (pun intended):
Somehow there was time and resources in the Mikdash to share the cups that caught the blood and send them down the line to the Mizbeach and back again to the next animal being slaughtered. During this time the knives could have been passed around also - and even checked for blemishes.
There were also enough bowls in the Mikdash that everybody could pour in the Eimurim that were eventually burnt on the Mizbeach, as well as supplying the poles (and some hooks) for everybody to skin their animals. But for some reason, everybody had to bring their own knife. Why?