Replace stair treads on a prefabricated staircase
I can't really tell what I'm seeing in your screen image, but I think your plan is reasonable. If the stringer isn't tall enough to add the individual 2x4 ledgers you describe, you could use steel angles bolted to it.

Were it my project, though, I'd be cutting the treads out and building three new stringers, using the existing setup as a pattern and adjusting for any variation in tread thickness. I'd cut them out of 2x12 or LVL stock and fasten them in place, using construction adhesive at all joints to add stiffness and prevent noise. It's not difficult and then you have a staircase built to high quality standards.