Never mind what gender the Dr is, let's consider the really important  thing. Hair colour! Gingers get a bad press, we've been blamed for all sorts of stuff, from Judas Iscariot onwards. Dr. Who would make a great role model for put upon copperknobs. and would lead to greater acceptance of the red headed!
Jodie Whittaker changes her hair colour at the drop of a sonic screwdriver, so why not a ginger Doctor?