I think you gave a good summery of the places to see. We did a twelve day tour in October 2008 and still had sweltering days around 40c. We had a float in the Dead Sea, just don’t get salt in your eyes. Masada; which you go up on a cable car; was the site hundreds of Jews committed suicide; rather than fall into the hands of the Romans, worth a trip if you can.
Someone in our group described Jerusalem as a religious theme park. You could queue up for an hour to get into many of the holy shrines. Once you were at the place you wanted to be, there was little time for reflection and prayer, because there was an hours queue behind you.
For me the most profound place I visited was the Dome of the Rock. It is said to be the most sacred place on Earth; where all the prayers of all religions go to, before they go up to God. This is the site of the original Temple built on Mount Moriah, where Abraham was commanded to sacrifice his son, it is where the first and second Temple were built.
When you walk in the central parts of Jerusalem, there are wells that are maybe 3 -4 meters deep, this is were the road was in the time of Christ. They have uncovered some shops which again were about 3 - 4 meters below the current road level. Many of the Christian sites are inside the Muslim area of the old city, this is surrounded by the Jews, we did get searched by police a couple of times.
Our Jewish guide asked us to be philosophical about the places we visited. She showed us two places where the last supper may have happened. One was an upper room above the tomb of King David, because Jesus was of the line of David. The second place was a cellar. This made sense because it would have been about the height of an upper room two thousand years ago, you take your pick.
There were two possible sites for the tomb of Jesus, one was the Holy Sepulchre. The other was the garden tomb in the side of rocks which seem to match the Biblical description more accurately. On some sites there was a church. mosque and a synagogue, linking the three Abrahamic religions to an event.
Mount Tabor, the place of the Transfiguration is worth seeing. You need nerves of steel if you go up in a taxi. The drivers think it is fun to go round hair pin bends with one hand on the wheel, whilst they turn round and chat to you in the back seat. I found the villages around Galilee to be a much more peaceful and reflective places to visit.
I first went to Jerusalem in 1965, when much of the city was within the borders of Jordan. It was at a much slower pace, and we had time to reflect and pray, but sadly I did not understand much at the time.