That's the part that's a myth: that it was widespread or normal. Of course small errors would happen.
But the main reason it didn't happen frequently is that employees at Ellis Island weren't even recording anyone's name. They were just checking them against manifests that were written elsewhere.
And when passengers were consulted about immigration-related issues, it was through an interpreter who would know the script and language of the arriving passenger.
More info: https://www.nypl.org/blog/2013/07/02/name-changes-ellis-isla...
How do you transliterate Жижщенко? Anything with a ё in it? Or an ы, or й? Was there a policy on differentiating between "е" and "э"?
While many of these immigrants wouldn't have been fluent in English, they still would have had access in their home countries or through the shipping company to someone bilingual who knew English as well. It's not like English was some incredibly inaccessible language to someone in Poland or Russia.