It seems very strange to me that a “king” would do the laborious and dangerous task of a breaker.

That would make for a rather unfortunate comment on Jesus as king! It is precisely the responsibility of a king to ‘judge us and go out before us and fight our battles’ (1 Sam. 8:20).

It still seems unlikely to me that the participle in Micah 2:13 ‘the one breaking through’ (הפרץ) refers to a significant figure other than one of the subjects explicitly mentioned in the paragraph - either one of the sheep, YHWH, or the king.

In the LXX there is even less reason to think that two figures are intended: we simply have the sheep breaking through a breach.

Yes, Jesus regarded himself as Israel’s true shepherd, but the passages you cite are too remote from Matt. 11:12 to function as an interpretive context. In fact, the thought of restoration does not seem relevant for Matthew 11, which has to do with the significance of John’s imprisonment and his preaching of a message that opposed the powerful and unjust in Israel. Notice that the quotation from Malachi 3:1 in Matthew 11:10 evokes the thought of God coming to judge his people - the image of sheep happily running out to eat grass is quite out of place in this context.

Nor have you addressed the question of how Jesus’ words would match up against the image of Micah 2:12-13:

From the days of John the Baptist until now the kingdom of heaven has forced its way (middle) or has been subjected to force (passive), and forceful men have seized it.