Leviticus 16 is completely non-sequitur.
God does also refer to the nations (non-Jews) as “My people” as referenced by Isaiah himself in Chapter 19:24-25.
“In that day shall Israel be the third with Egypt and with Assyria, even a blessing in the midst of the land: Whom the Lord of Hosts shall bless, saying, ‘Blessed be Egypt My people, and Assyria the work of My hands, and Israel Mine inheritance.’”
Which is also absurd, for what Is. 53 describes is what Lv. 16 requires, and if Israel is to be the atonement then it must fulfill what Lv. 16 describes, and under which rubric other provisions for atonement are made.
God does also refer to the nations (non-Jews) as “My people” as referenced by Isaiah himself in Chapter 19:24-25.
What is in the prophetic sense, and requires the unblemished atonement of Lv. 16, which Israel certainly was not, nor does Israel overall correspond to the descriptions of the suffering servant in Is. 53, despite the stretching of imaginations. Thus rather than Is. 53 making Israel the atonement he is in need of the atonement Lv. 16 requires.
Thus you have ancient rabbis who saw Is. 53 as referring to the Messiah.