Consider a "standard" C channel for a emergency exit stair going the same net rise and run: There, for a (somewhat smaller) stair tread width, you need a 1-1/2 or 2" "flange" (missing from your flat bar) and a 8-10 inch deep web stiffened at both ends with 1/4 thick steel.
Just a flat bar 1x12 seems pretty reasonable in comparision, doesn't it?
Try this: If your building code allows it, add a center flange between the two outside stringers. That will let you reduce the depth of the outside stringers slightly.
Second, carry the 1x12 plate up past the ends of the stair trends to form an archt. feature that reduces the apparent height of the outside plate. The stair treads will go into the "Side" of the stringer plate, rather than going over the top of the plate, but the higher side plate merely serves as a toe plate instead. This may be your intention already.
When I roll square tube steel for decorative swept/spiral iron stairs, I need to create three bends simultaneously: The obvious one is in the "plan view": the stair curves in a circle. Superimposed on this is a spiral as the stringers go down: If the rails and horizontal members are round (1-1/4 Sch 40 pipes are common) you can roll the pipes in both curves at once and get a good aproximation of a fit.
If the stairs are swept vice a constant radius spiral, then each sweeping section has to be templated and rolled manually. Stick to no more than 10 foot sections in any case. That way, if the roll screws up, the previous and next 10 foot section can still be used. Weld, and grind smooth, the final stringers to get a look of one smooth piece of steel.
But, if the rails and horizontal members are rectangular - as they are here, you will need to "twist" the member to keep its top "flat" and its sides "vertical" as they go down the spiral. The mental image of a screw threads is a bit mispleading because an actual screw thread isn't "square" - it is too small to see the twisting effect at that scale anyway. Try a couple of flexible (plastic) 1/4 x 2 trim molding pieces from Home Depot, at home bend them into a spiral approximating the scale stairs you have.
Two options: I usually use 1x1 11 GA tube steel for residential iron stairs, most of the time these are less than a slope length of 14-15 feet so I roll two 8 ft sections, cutting off the 12" waste at each end of the rolled piece. (The rolling machine can't bend the last 8" of the member.) A commercial building is going to be longer (higher floor-floor heights) and with shallow stair rise/run ratios.
Once you are comfortable with the look and size you need, and 1x12 isn't too conservative, but it is heavy for a 43" some-odd wide inch stair, dimension and draw the sides as you want them. The fabricator then will have the (uneviable) job of rolling, heating, jigging, templating, and twisting that piece of iron. Unfortunately, he will have to do two different setups: since the inner rail is substaintially different than the outer.
Before you hire ANYONE to roll (and twist) your steel, get photos (or personal vists) to at least 2 of the bidder's existing stairs. Few can get these kind of steel stairs right their first time, and errors are ugly. Of course, that might be the owner/contractor problem - not yours as designer 8<)
Robert