Ta Daaa, I am quite familiar with San Jose, and although we do not have a frost line out here, many but not all parts of San Jose have highly expansive soils, with heavy concentrations of clay and silt, which can and do create issues with settling and heaving of paving and footings. That being said, there are many solutions, from isolated islands, that either are mounted on heavy duty permanent footings to mobile units using heavy duty casters. We also have light weight aggregate stones fascias that can allow for less settlement issues. All that being said...
If it were me, I would take up some of the pavers, they simply will not support a heavy structure and I would go with concrete masonry units over either wood, steel or aluminum. Even though professionally installed pavers would be more stable than your typical homeowners work, they will still heave, or settle, if you are on clay. Now. there are also gravel substrates in San Jose, those are extremely stable, and that would allow for placing a lighter wieght CMU construction on the pavers.
I do not like Pressure treated wood for exterior carpentry, as any cut exposes the core of the wood, which, typically is not as treated. Further, recent changes to the chemicals and processs of pressure treating leaves most P.T. wood more suscpetible to rot, my experience has been that it will last less than 10 years. Aluminum is cheap, but, is subject to corrosion just like any metal, but, it fails catastrophically, which is to say, one day it looks fine, the next day it fails. Galvanized structural steel is a fine material, and one that if I was building a house, I would not hesitate to use. However, once you cut it, the edges are subject to rust.
BTW, there are ways to frame a structure and isolate the backer board for the stone fascia in such a manner as to prevent expansion of the wood structure from the fascia. However, a CMU substrate will be much more solid and durable. If it was me, I would do the exterior three wall with CMU, and use easily replaceable components to 'frame' out the interior. This way, the overall appearance will remain consistent and attractive while the components can be easily repaired. And I would probably use steel with painted cuts.