Can I build a mitered aluminum glass frame from glass edge banding?
What you describe as to structure can be built if you have:


A source for the aluminum extrusion frame material. If no source you may need to design its cross section profile yourself and then finance a manufacturer to extrude the material for you. 
A source for the corner brace piece. For a custom designed extrusion you would need to design around some existing corner or design and get custom corner pieces stamped out. Be prepared for a tooling charge for custom parts.
The tools needed to cut the aluminum extrusions and a press to push in the corner brackets. Depending upon the method used to retain the corner pieces you may also need  a tool to crimp or rivet the part in place.
Patience to put it together with the glass panels.


What is not at all certain is if this will provide the protection against glass breakage and shattering that you desire. Obviously the design of the frame will need a good bit of engineering modeling to fully understand the rigidity and strength of the frame to protect the glass. It is almost assured the slimmer, thinner and lower profile the frame the less protection it will offer to the glass. There will also be considerations on the type of glass used, its panel size and thickness. 

Another factor that also plays into the protection level is if the frame will include a rubber or vinyl isolation strip between the frame and the glass. This alone will likely require considerable engineering work and experimentation.