You can do what you want, even if they are housed but you will need to loose the ceiling material or move the stair treads ahead about 6". you could easily do that on the short lower run, but not so on the upper run.
You will need access to the upstairs after the stairs are gone, this may take 2 days to do, especially if you are a novice, maybe even 3. Seasoned carpenters really should be doing the work.
If the lowering of the ceiling is a deal breaker for doing it, no need for me to write anymore. I see no issue with lowering the ceiling 4 or 5", the square corner that projects below the sloped section in one of your pictures will still be lower than the new dropped height of the stairwell. If you feel you have enough experience to tackle this, please say so and I will edit the post if need be, but it is pretty much a gut and replace. Leaving anything in of the original will only slow down the new install or weaken what you add over it.
There is really no good way to patch this up or fix it without access to the area under the cracked tread. So I would suggest just doing that - open up the stucco covering under the subject area so there is enough room to work in there.
The nature of the fix will depend to some degree as to weather the crack is open or tight. Since you say that it is barely noticeable lets say that the crack is tight and you would be satisfied to just leave the crack width as is and just solidify the tread. Since the underside of the tread is probably unfinished you get an excellent opportunity to be able to use wood glue in the repair process. If there is paint, finish or stain on the underside of the tread you would have to sand it off so that you can repair with glue. The best glue is the yellow variety of professional wood workers glue.
To apply a fix you will want to cut a piece of wood that will fit comfortably under the tread up against its bottom side. It does not need to cover the whole bottom though. Make sure it bridges the crack by at least 2 inches on either side and is long enough to extend from the end with the crack to well past the other end of the crack. I would use a hardwood board for this patch (oak, maple or birch would be good ones to use).
You want to prepare this add in board for screwing to the underside of the tread. The best screws are flat head wood screws with Philips type slots (instead of straight slot screws). Make sure that the screws are the correct length so that they go through the thickness of the add in board and into the tread without going through the tread. If the patch board is 3/4" thick and the tread is 1" thick then a good screw length would be 1 1/4". Pre-drill clearance holes through the add in board along its sides so that they are away from the crack area. Counter sink the clearance holes just enough so that the flat head screws are just even with the surface of the board. Next you will need to drill pilot holes up under the tread in the correct locations so that screws can go into the tread without splitting it more. A good way to do this is to hold the prepared add in board in place and drill through the clearance holes with the pilot bit up into the tread. Use a stopper on the drill bit so that you do not accidentally drill all the way through the top side of the tread.
Next apply the yellow carpenters glue to the side of the add in board opposite the counter sink holes. :-) Spread the glue all over but do not leave an excess either. You could also apply some glue to the underside of the tread since it is common to apply glue to both sides of a joint but you can skip that if you have placed just the right amount on the add in board.
Then you put the board in place and install the screws into the pilot holes and firmly tighten into place. Be careful not to over tighten and strip out the screws in the pilot holes. Since the screws are only going into the treads about 1/2 inch they could strip out of you get two aggressive. I recommend hand driving as opposed to using a power driver for just this reason.
When it comes to patching the stucco area you may decide to take a more drastic step and remove all the stucco and re-do the area with drywall. It is much easier to work with and patch up if you ever have to go through it again in the future.
Best Answer
I repaired my poorly-build stairs in just this manner recently. Here's what I did after removing the drywall:
Run a bead of high quality construction adhesive along all joints and laps. Every point of contact between stringers, risers, and treads should have one or two beads, depending on access. Use a protected finger or tool to press the bead into the corners and gaps, spreading it and enhancing contact.
If the stairs are at all springy, add 2x4 or 2x6 lumber to the stringers. You could even add two members to the center stringer. Use construction adhesive and 2-1/2" gold construction screws.
The result is a stone-solid staircase with no noise whatsoever. If you have any noises remaining after allowing a night for the glue to set up, tackle them on an individual basis. Chances are more glue and maybe some shims will do the trick.
Reinstalling drywall on the underside is trivial for your hangers. I doubt that they'd even upcharge for it, especially if they bid to tape that area anyway.