I know there’s a million stories like this but no names were ever changed by officials at Ellis Island. There wasn’t even a procedure in which they could do so. (One brief source.) People often voluntarily changed their names to fit in, but claiming that they were forced to do so made put the blame elsewhere.
The equally simple reason for why actors today keep their names is because they can.
Besides the other good reasons listed, the classic studio system wasn’t far removed from the days when actors were the world’s trash and going into acting tarnished the family name. People changed their given names to hide their identities and protect their relatives. Probably not many people in western culture think that way any more.
People most certainly did get their names changed at Ellis Island. If a person who was illiterate from a Slavic nation gave their name as “KO-vach,” It usually got taken down and spelled “Kovatch,” or sometimes “Kovach.” If the person was literate, they wrote it as Kovac, Kowacz, or one of a few other ways it might be spelled depending on whether they were Polish, Czech, Croatian, or a few other ethnicities that had the name, and the Ellis Island scribes ignored diacritics. So you have “Kovac,” pronounced “KO-vac” instead of “KO-vach,” “KO-wash,” instead of “KO-vach,” and other people who spell their name Kovacs, but insist it’s Kovach. Some people end up with hybrids, like “Kowach.”
Some people who spoke English chose to answer the question “What is you name?” By giving an English translation of the name. Some did not. We have a branch of the family who came from England, and chose to answer the "What is your name question as ‘Black’ instead of ‘Schwartz.’ They spoke The Queen’s English, traveled second class, settled in Boston, and just quietly started attending the Episcopal church.
Honestly, it’s a myth. It happened exactly once, but that was a weird thing. I know people want to believe in the myth, but every authority on the subject can give detailed accounting on why it’s a myth.
Albert Einstein uses his mother’s maiden name, Finney, as a stage name for obvious reasons.
Laura Jeanne Reese Witherspoon probably goes by Laura Toth when she’s doing the IRL thing.
I can’t blame Rush guitarist Aleksandr Zivojinovic (pronounced (zi-vee-no-vitch) for Anglicizing his name. His Yugoslavian-immigrant father actually considered changing the family name to “Lifeson” long before his son ever entertained the thought of a musical career.
At that, Oscar Isaac has explained how, “starting out as an actor, you immediately worry about being pigeonholed or typecast. I don’t want to just go up for the dead body, the gangster, the bandolero, whatever. I don’t want to be defined by someone else’s idea of what an Oscar Hernández should be playing.”
It depends on what you call a “stage name”. To join SAG-AFTRA you need a non-conflicting unique professional name. It may just be that people tend towards a first and last name that make it appear to be uncommon but it is still typically the norm.