I am not 100% up to speed on the variants issue, but I believe that the cornerstone of the Bible's veracity is that the Scriptures are so consistent over the span of thousands of years and multiple authors, testifying to the inspiration of the HS.
My summization on the variants is that, while variants exist with differing degrees of "variation", they still do not detract from overall orthodoxy, thus the HS inspired message is not impacted by them and this fact substantiates the infallibility of the words from the HS.
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Regarding the evidence, suffice it to say that significant textual variants that alter core doctrines of the NT have not yet been produced.
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