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 tar

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.