Why would Lot offer up his two daughters to the evil crowd in Sodom?
Thanks for the question. It's a good one!
I think this was a function of the misogynistic thinking of the day. See Judges 19 for an almost identical but much more tragic example of the same kind of thinking. Note that chapter 19 begins thus:
Judges 19:1 And it came to pass in those days, when there was no king in Israel…. which, I think is an “Ellipsis” for the complete phrase which occurs in chapters 17 and 25.
Note that before and then after this incident in Judges 17:6 and 25:21 we read: In those days there was no king in Israel; everyone did what was right in his own eyes.
By "no king" I think the real import is that they did not base their behaviour on the Word/Law of their true king, God:
1 Samuel 8:7 And the LORD said to Samuel, "Heed the voice of the people in all that they say to you; for they have not rejected you, but they have rejected Me, that I should not reign over them.
Lot was influenced by the thinking of the world of his day just as was the Levite living in Ephraim. I think that this is a horrible example of the damage that worldly thinking does to the attitudes of those who claim to follow God. Lot and even the Levite, who should have known better, did not seek God's guidance in this but "did what was right in their own eyes."
It's interesting that the angels didn't even deign to comment on Lot's suggestion of substituting his daughters for the two visitors but rather they just unceremoniously hauled him back into the house.
Once back in the “house” in the presence of God [the angels] he could be delivered while the children of darkness groped blindly for the door and could not find it.
There is no indication, either in Lot's case or that of the Levite's concubine that God sanctioned their actions. I think Lot's "offer" and the Levite's action are offered as stark examples of the horror to which the thinking of the flesh can cause even normally well-meaning people to sink. Overt condemnation would have been superfluous.
I hope that helps,