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Ah but at immigration you may have been expected to spell your name and were sort of at the whim of the official (who was helping intake many people)

Lots of reasons you might not have your official name written or heard correctly

I would assume the reason is not a legal one, but a cultural one, i.e. having a very obviously foreign name a few generations ago in the US might make you a target of discrimination. Changing something like "Alexei" to "Alex" would help you stand out less.

My great grandfather emigrated as a young person and started using his father's patronym as his own last name instead of continuing to use his father's first name as his patronym. He could have kept the patronym tradition alive, but why? He was in America now, and once his kids were born, they mostly spoke English anyway.