I think it was the other way around my father told me a story of feight being pronounced "fight" the names if you do a census search seems to be older "fite, fait (from French) feight" all seem to be older than Fate. which is an english word and how immigration authorities would probably write it down (didnt matter how you said you spelled your name they wrote it however they wanted.)

My grandfather Raymond Edward Fate (son of August Fate) was from Allegheny Pa near Pittsburgh.

Raymond James Fate