So, how do you explain diagonal wall bracing?
Diagonal bracing takes loads parallel to its (straight) long axis where it is strong. Stringers take loads in the other direction, where they are weak.
In this example here -> if the stringer was screwed to the wall with no stairs, then it would be all good - just like a diagonal brace.
But when a stringer is added on the other side, (so both stringers are supported differently from each other) the stair case cantilevers a bit to one side and the assembly will twist as load goes on it and the less supported stringer bends more. As the stair case twists, it will also twist the wall with it, slightly like a cork screw. (Not enough to see or feel much, but enough to weaken the wall from what it would be otherwise.)
I've built quite a few sets of stairs, always fastened the inboard side to the wall when we could. Usually put a 2x6 on the wall and then nailed the stringer to that so the drywallers could slip a simple angle-l cut sheet in behind the stringer and rough treads.
This is a useful arrangement and I've done similar installs before. Structurally, a more elegant way to do this would be to secure the 2X6 to the stringer (to pad it out) before putting up the stair case and not attach either to the wall.
Fastening one side doesn't make the stairs any weaker- it actually makes them stiffer since the entire structure is anchored by one fixed side.
Correct. It makes no difference for the stairs - but the wall gets weaker. If there is a situation where the stairs deflect and need an intermediate attachment, they should probably be supported on both sides.
They probably won't crash down but they will corkscrew. I know that for a fact, I read it on the internet.
Or, you can use calculus.
Hopefully it will be a "mild" Class 1 corkscrew event, not a "monster" Class 5 corkscrew episode that launches you out into space with a crash landing the middle of the garage floor. Either way, for safety's sake don't carry an actual corkscrew while ascending or descending those stairs. That would be like running with scissors.
You have been warned.
I mentioned below that this will probably not ever cause a real problem for most situations. This type of thing becomes worse than running with scissors in the extreme cases like hurricanes and earthquakes where the wall gets pushed to its limit.
If these details aren't important, then we can throw out every other hurricane strap, skip the plywood clips, pass on gluing down shingles and all that other stuff. What the hell - if this actually happens the homeowner won't be around to complain and we'll be long since gone.