I've found questions asking about an em dash in quoted speech, and punctuation rules for em dashes in quotes, but this question is quite different.
The example I saw comes from a tweet, reproduced here for convenience:
"For today in the city of David there has been born for you a Savior—not a divine social justice warrior, not a liberation theologian, not a critical race theorist, not an ecumenical pluralist, not an interfaith universalist—a Savior, who is Christ the Lord." — Luke 2:11 (NASB)
One reply tweet argued that that is not what that scripture says, obviously referring to the text between the first two em dashes. My question is, would this be a valid usage of em dashes, namely to signify an interjection within the quote, that is, text which is not part of the quote itself?
Edit: It has been pointed out that the use of square brackets is sometimes used in quotes to modify the quote for contextual clarity. See here for example. Could the em dashes in the example above be replaced with square brackets to achieve the desired effect? Maybe. But this deosn't answer the question of whether em dashes can be used correctly to perform this parenthetical/interjection function.