I posed this question some 20 months ago and have puzzled over it many times. It always seemed bent and I found no satisfying answer. For some reason, I didn't think to check how Ezekiel 9:6 read in the Greek text. When I did, it became, I believe, very clear:
[Brenton Septuagint Eze 9:6] Slay utterly old man and youth, and virgin, and infants, and women: but go ye not nigh any on whom is the mark: begin at my sanctuary. So they began with the elder men () who were within in the house.
[Eze 9:6 LXX] (6) πρεσβύτερον καὶ νεανίσκον καὶ παρθένον καὶ νήπια καὶ γυναῗκας ἀποκτείνατε εἰς ἐξάλειψιν ἐπὶ δὲ πάντας ἐφ᾽ οὕς ἐστιν τὸ σημεῗον μὴ ἐγγίσητε καὶ ἀπὸ τῶν ἁγίων μου ἄρξασθε καὶ ἤρξαντο ἀπὸ τῶν ἀνδρῶν τῶν πρεσβυτέρων οἳ ἦσαν ἔσω ἐν τῷ οἴκῳ
In 1 Peter, Peter identifies himself as an elder, and in the next chapter he specifies that he is exhorting the elders:
[1Pe 5:1-4 CSB] (1) I exhort (παρακαλῶ, present active indicative) the elders among you as a fellow elder and witness to the sufferings of Christ, as well as one who shares in the glory about to be revealed: (2) Shepherd God's flock among you, not overseeing out of compulsion but willingly, as God would have you; not out of greed for money but eagerly; (3) not lording it over those entrusted to you, but being examples to the flock. (4) And when the chief Shepherd appears, you will receive the unfading crown of glory.
So it seems clear that Peter understands Ezekiel 9:6 LXX to be saying that judgment must begin in Jerusalem, with the leaders of God's flock. He is referring to the horrors and destruction of the temple, Jerusalem, the Jewish polity of the Great Jewish Revolt c. 70 AD/CE.
In the gospels, Jesus tells parables to the effect that he is going to judge the leaders of the temple based polity (the scribes, Pharisees, the Sanhedrin, etc.) for their leadership as well per Ezekiel 34. Please see "the sheep and the goats" in Matthew 25 and:
[Mat 21:43-45 NKJV] (43) "Therefore I say to you, the kingdom of God will be taken from you and given to a nation bearing the fruits of it. (44) "And whoever falls on this stone will be broken; but on whomever it falls, it will grind him to powder." (45) Now when the chief priests and Pharisees heard His parables, they perceived that He was speaking of them.
[Jer 25:29 NLT] (29) I have begun to punish Jerusalem, the city that bears my name. Now should I let you go unpunished? No, you will not escape disaster. I will call for war against all the nations of the earth ["land", of Israel]. I, the LORD of Heaven's Armies, have spoken!'
[Eze 9:6 NLT] (6) Kill them all--old and young, girls and women and little children. But do not touch anyone with the mark. Begin right here at the Temple." So they began by killing the seventy leaders.
[Mat 3:10 NLT] (10) Even now the ax of God's judgment is poised, ready to sever the roots of the trees. Yes, every tree that does not produce good fruit will be chopped down and thrown into the fire. [Note: I understand "trees" in scripture to refer to teachers.]
[Luk 12:47-48 NLT] (47) "And a servant who knows what the master wants, but isn't prepared and doesn't carry out those instructions, will be severely punished. (48) But someone who does not know, and then does something wrong, will be punished only lightly. When someone has been given much, much will be required in return; and when someone has been entrusted with much, even more will be required.
[Eze 34:17-24 NLT] (17) "And as for you, my flock, this is what the Sovereign LORD says to his people: I will judge between one animal of the flock and another, separating the sheep from the goats. (18) Isn't it enough for you to keep the best of the pastures for yourselves? Must you also trample down the rest? Isn't it enough for you to drink clear water for yourselves? Must you also muddy the rest with your feet? (19) Why must my flock eat what you have trampled down and drink water you have fouled? (20) "Therefore, this is what the Sovereign LORD says: I will surely judge between the fat sheep and the scrawny sheep. (21) For you fat sheep pushed and butted and crowded my sick and hungry flock until you scattered them to distant lands. (22) So I will rescue my flock, and they will no longer be abused. I will judge between one animal of the flock and another. (23) And I will set over them one shepherd, my servant David. He will feed them and be a shepherd to them. (24) And I, the LORD, will be their God, and my servant David will be a prince among my people. I, the LORD, have spoken!
[Luk 21:20 NLT] (20) "And when you see Jerusalem surrounded by armies, then you will know that the time of its destruction has arrived.