Because Danzo was intent on taking his other eye, killing him and it certainly would have happened. He was dead anyway, he knew he could not beat Danzo so he just committed suicide. Gave eye to Itachi so Danzo could not have.
Itachi suggested the eye's KA couldn't be used at that point and that's why he didn't use it on Sasuke & instead gave it to Naruto to use on Sasuke much later when it had fully recharged.
Danzo just so happened to take the only fully charged Kotoamatsukami eye.
plus Danzo would know where he is headed and may have had more reinforcements in place near Fugaku's area.
Because both eye's are needed for Tsukuyomi to work. So needed danzo to believe he was dead (taking his eye with him)so that the plan wasn't going to be complete, thus leaving itachi(shisuis partner) undetected to stop the THE uchiha plan to attack the leaf and keep danzo from getting his other eye.