My father was telling me that immigrants USED to anglicanize their names. For example, an uncle of mine had a very Italian name and he changed it to Jim when he immigrated.
But it's not just anglican names people used to change the spelling at AT LEAST the pronounciation to make it easier. And if they wanted to name their child a cultural or family name, they would give them an American name to go by. I actually have a friend who is Greek and her real first name is Greek. And like many Greek names it sounds very pretty but is particularly difficult to say. So her mother gave her an Anglican middle name. That's assimiliating.
Now people want to stand out.
It's practical. All of the different one-syllable names and different spellings of the one-syllable names are just impossible to remember for a Westerner.