Believe your contractor. He is the pro and should know what he is doing. Wood flooring changes direction all the time.

Run the flooring in the living room and family room the same. Since the entrance from one room to another is wide, they should run the same way and squared up with the walls. The flooring could run lengthwise or across the rooms. But, take your cue from the cabinets on either side of the fireplace and continue that line throughout. Diagonally gets busy and, judging by your post above, it looks like you prefer a more organized setting. In the pix below, the flooring is having a battle with the squares in the doors of the cabinets.

Keep the flooring beautiful but in a simple way so that you can focus on the rest of the things in the room. It will work just fine and make sense if you do the 2 spaces the same and if the woods and stain are a match. You will forget about it in a short while and, once you move back into the rooms with all of the furniture, rugs, etc., it will look like it was always there.