RE: "If the name on the manifest is the name they used..."
That is not the gist of what has been said... The name on the manifest was the copied from the name on the embarkation manifest.
RE: "I can decipher some, but nowhere near all, the names."
There are instances of poor penmanship. But no matter how bad the penmanship of someone who transcribes it, I still know my name. The same was true for each immigrant.
RE: "But the names were never changed."
There are instances of immigrants who changed their names, but the ship manifests were not the place where the changes happened... although in some cases, an immigrant may have decided to adopt an inadvertent misspelling.