Nothing wrong with getting new handles from GrillGiant,
but, IMO, when it comes to the twin rivet handles, try to save them if possible.
I restored this one last summer.
Using pretty much the advice mentioned above.
I gave a light sanding with 320 grit sandpaper, then 0000 steel wool.
I then applied 1 or 2 coats of tinted Danish oil, which is more-or-less, tung oil with pigment.
Then a couple three coats of Helmsman spar urethane (satin finish) with a light sanding between coats.
It has held up remarkably well considering I left this kettle out all winter, and the handle looks great!
EDIT: I did not need to remove the handle.