That would work. You can use a degreaser like gunk or jizer on the greasy parts. Test a small area first. Both are water soluble, for best results wash off with hot soapy water. WD-40 and tin foil on chrome parts. Sovol metal polish for the alloy and good quality wax for the paint work. You may need to use scotch brite pads on the alloy and then metal polish.
Don't mirror polish alloy stuff, as it will look odd with the patina of the frame. This has the potential to be something really beautiful with care and time. Good luck

Edit WD-40 is good for cleaning and getting rid of sticky residue.