Keep safety and following local building code as your priorities.
It seems like what you want is a pony wall with a bookcase built in. Installing a wall that size would be relatively easy, but you may want to hire it out. You’ll also want to check if you need a permit.
Here is a post someone made about their project switching from balusters to a pony wall. (I only skimmed the post, so follow their instructions at your own risk!)
You could also cover the metal railing with something like beadboard paneling inserts, though it would take some precision cuts to accommodate the newel shapes. That would essentially make a faux wall without removing any of the integrity of the current railing. Then just find the shelves/cabinets you like and place them on the floor. Add trim etc as desired and paint the whole thing one color to give it a built in look.
Keep in mind that the balusters let a lot of light into the stairway. The stairs will feel much more tunnel like if you create a solid wall there.
In the meantime install a mesh child safety guard so your kids don’t slip through.