Do stair stringers need to be supported along their length?
Stair stringers are normally designed to be supported at the top and bottom, with no further support needed.

I can't say for sure whether the framing that is there currently is to support the stringer, but I would expect that it's just there to support the drywall.

Read up on how to tell if a stud wall is loadbearing or not, if it's not loadbearing, then the current framing is just for drywall.