Properly constructed stud partition walls should not be "weak and wonky".
Framwork stud wall should be constucted with 3x2 CRS timber (minimum) with horizontal inter-bracing for strength and support. Soundproofing material should be fitted between the stud wall timbers for soundproofing. Exterior plaster board cladding should be 9.5mm thick minimum...ideally 12mm. Quick and easy to construct in no time. I suspect you don't have a wall as described above but something else. Mel.
I'm sure you're right that it's not constructed as you describe. Presumably it could be skimmed with plaster, or rebuilt as you describe. Any idea how much it costs?