Jesus birth location?

These were my two friends on the walk from Jerusalem to Bethlehem. This cave had an interesting construction. At the immediate entrance to the cave there was a low type of enclosure. It appeared that the front part of the cave was used for the animals. Then as we proceeded back into the cave, there was another higher barrier, that seemed to set off the living space of the family from the area prepared for the animals. In this living space there was a hole in the ceiling of the cave, through which we could see the sky above. It was clearly marked with black from the fire that would warm this area. The hole allowed the smoke to exit the cave.
One of my favorite scripture scholars is Kenneth E. Bailey. In 2008 he published JESUS THROUGH MIDDLE EASTERN EYES (Cultural Studies in the Gospels). The first chapter is titled: "The Story of Jesus' Birth" He states: "The traditional understanding of the account in Luke 2:1-18 contains a number of critical flaws." He then lists five critical flaws.

But the following text together with an accompanying diagram evoked my experience of traveling to Bethlehem, page 28,29: "For the Western mind the word manger invokes the words stable or barn. But in traditional Middle Eastern villages this is not the case. In the parable of the rich fool (Lk. 12:13-21) there is mention of "storehouses" but not barns. People of great wealth would naturally have had separate quarters for animals. But simple village homes in Palestine often had but two rooms. One was exclusively for guests. That room could be attached to the end of the house or be a "prophet's chamber" on the roof, as in the story of Elijah (1 Kings 17:19). The main room was a "family room" where the entire family cooked, ate, slept and lived. The end of the room next to the door, was either a few feet lower than the rest of the floor or blocked off with heavy timbers. Each night into that designated area, the family cow, donkey and a few sheep would be driven. And every morning those same animals were taken out and tied up in the courtyard of the house. The animal stall would then be cleaned for the day. Such simple homes can be traced from the time of David up to the middle of the twentieth century. I have seen them both in Upper Galilee and in Bethlehem." It surely seemed logical to me that the structure of the cave that we had witnessed along our hike would be transferred to a building at a later time.
Bailey goes on to explain the Greek word katalyma. NRSV (Lk. 2:7)translates this word: "...there was no place for them in the inn." Bailey goes on to tell us that Luke used this word on one other occasion in his Gospel (Luke 22: 10-12) "The teacher asks you, 'Where is the guest room (katalyma) where I may eat the Passover with my disciples?'"