Amazing. No sooner than the "community" discards the canard that immigration officials changes people's names, another canard that name changes were effected by HIAS officials. Wow. That is even more imaginative and less plausible.
Here's what happened, again and again: immigrants adopted new names, willingly and deliberately. The reasons for doing so varied – some names were in foreign alphabets (e.g. Cyrillic), some name spellings were not pronounced properly in English (e.g., Romanian -vici suffix sounds like "English' -witz), some names were deemed too long to be practical, etc., etc., etc. But no reason was needed to use a new spelling or a new name. And in many (most?) instances, no government official or other representative was required to sanction the name change. Google the history of name changes in the U.S., and that is what you will find.
It was also possible to effect a name change at the time of naturalization, but again, this was done willingly and deliberately by the person being naturalized.
Were mistakes made, in misspellings, or later imputed to bad handwriting? Sure, but again, I believe these errors did not become "memorialized" unless the immigrant adopted the mistakes or misspellings, willingly and deliberately.
BTW HIAS is still around, and available for inquiries. Good luck chasing down this new theory.