Naturally dry scalp (xerosis) is not that common as a cause of dandruff or flaking, more often people self diagnose and actually have seborrhoeic dermatitis, irritant contact dermatitis, atopic eczema or psoriasis. SD and psoriasis can benefit from treatment with SA.
I would not have thought there would be that high a concentration of salicylic acid in a cosmetic shampoo, and you can always dilute down before use. Yes SA can be drying because it is fat soluble, but how much would depend on the formula - if there is oil in the product the SA has 'options'. It's also an acid so will help close the cuticle and a keratolytic (breaks down protein bonds) so would presumably damage hair at higher concentrations. IIRC it's generally at 3% in pharmacy type shampoos, up to 10% in skin only topicals.