What problems will I face with this all glass stairwell?
I live in a mild climate and we designed a house for a client with a large greenhouse on it facing south. 

We installed thermostatically controlled exhaust vents and the space still stayed extremely hot. The client was nice and she found it perfect for growing oranges, tomatoes, etc. year round. 

I’d be concerned about extreme heat build up and put your architect on notice that it must be comfortable without running AC constantly. 

Also, some window glazing seals are only rated to certain temperatures. If it gets too hot, the seals could fail. 

Also, you don’t want metal handrails (or anything you could touch) because the metal could overheat and be too hot to use...especially with direct sun hitting the handrail.