There are several books written on stair building, have you seen any of those?

No but I can check one out

I'm not seeing anything that needs additional support.

Ok wasn’t sure if over time the stairs would bow or not.

In many cases stairs enclosed on both sides or have a half-wall or railing if the basement is finished.

In older homes, staircases open most of the time, and now when the house is being sold they must be enclosed or framed to comply with the code and have a graspable handrail... without that they will not issue a certificate of occupancy.

Here is an example we just had to do in one of the houses that failed inspection.