Open Riser Stairs / Removing carpeting and converting to hardwood
I'll try to address your questions, though they're still a bit nebulous.
Is this an opportunity to replace the treads with solid hardwood?
Sure. Why not? As far as I'm concerned, a weekend presents such an opportunity.
Is this a conventional project?
No. Why does that matter?
The reason may be that there aren't many open-riser stairs to begin with, and it's not often that stair treads get replaced in general. That doesn't mean you can't or shouldn't.
What is the conventional method of covering the underside of the tread?
I'm not quite sure what this means. If you're installing wood treads they'll be wood on the bottom, too. If you're wondering about the steel brackets, I'd paint them or have them powder-coated, maybe. Stainless might be another option if you want high-class. You'd have to have a machine shop cut sections from angle stock and drill for bolts. I might invert them for an exposed style and so you can fit the treads well to the stringer/skirt. Speaking of, obviously you'll need to strip and refinish that.
Is adding 1/2 inch to the top of the tread going to make the first step noticeably rise too high?
Well, it would, but you're reworking your mounts anyway, so you can set them to the proper height.
A few other thoughts...
Tread options that come to mind:

Solid oak (not really that expensive if you buy them from a big box store)
Bullnosed construction-grade pine/fir lumber (2x10 or 12)--I've built such a set in my own home and it was beautiful

Some general notes:

Wooden treads can be slippery; consider placing silica grit in the upper varnish or wrapping them with a thin runner
Open-riser stairs are actually illegal in many cases nowdays for safety reasons--more than a few broken legs have resulted from slips through the void
Recognize that you have fewer stringers than modern stairs; be aware of how much strength and stiffness you'll need in a tread for safety and a quality feel