Yes it is silent. Cook and Cooke sound the same and when the name was recorded for births, deaths, marriages etc, the person recording it often made assumptions in the context of spelling.
Willet, Willett, Willitt etc may have suffered similar 'garbling in transmission'.
Willet is, I see, is an archaic form of Will(iam).
The point is, I suppose, do not get too worried about spellings and there is almost infinite variety.