I have done the same as you described with my new 64 GB iPhone 5s. I was scheduled to receive a replacement but some "issues" came up. I have done the restores as you mentioned you did and they take hours to complete.
One thing that I did note, to reduce the number of crashes was using WiFi whenever possible. I have the ATT 3G Microcell running at home with a cable internet connection.
As much as I like LTE, I think it is tied directly to the crashes and the chipset that is in the iphone 5s. Upon receiving the new phone in the first month I used LTE whenever possible. It was great until ATT started to throttle me at about 5 GB of my "unlimited" data plan. I continued to use it until the new biling period taking my usage to 7 GB for the month. I typically in the past never used more than a max of 2 GB od data on any iphone in the past.
Long story short this has helped with the crashed, but it came at my expense, I had to add Cable Internet connectivity from Comcast as the ATT DSL just did not offer enough speed and a consistent connection. The worst part of being throttled by ATT was being in need of some decent data speeds when away from a wifi location and not having it, they throttled me to a point that my phone's data barely worked over the cellular network.
Last month was my 2nd month of having the 5s. I used about 560 MB of cellular data and had less than ten full crashes of the iPhone. But there again I was on WiFi 99% of the time. Strange??
What is the memory capacity of your 5s?
How many apps do you have on your phone (a topic the tech support reps like to bring up, but should have not bearing on the crashes)?
What are you going to do next?
Did the tech support reps offer you any other solutions?
I think this could very well be an Apple issue that is either a problem the chipset or the software or both. If you Google this topic you will get many hits.