No it does not make any such claim!
Here's a direct quote from the footnote on page 1393 of my own hard copy of the New King James....
"Verses 9-20 are not found in the two most ancient manuscripts, the Sinaiticus and Vaticanus; others have them with partial omissions and variations. But the passage is quoted by Irenaeus and Hippolytus in the second and third century." - Footnote for Mark 16:9 - New King James Bible."
That footnote is 100% factually true! And, not only that but Nelson Publishing, the publishers of the New King James, agreed with you and included the verses in the text!
Now when it comes to the online version, your confusion is more understandable but still incorrect. Here's the footnote as it appears on BibleGateway.com....
Mark 16:9 Vv.
9–20 are bracketed in NU as not in the original text. They are lacking in Codex Sinaiticus and Codex Vaticanus, although nearly all other mss. of Mark contain them.
Note that is DOES NOT make the claim that verses 9-20 are not in the original! It states CORRECTLY that the NU omits the verses as not being in the original. Guess what?! The NU is NOT the NKJV!
What is the NU?
Well, I didn't know either, but, instead of assuming, I spent about thirty seconds and looked it up. The New King James itself defines it...
NU-Text
These variations from the traditional text generally represent the Alexandrian or Egyptian type of text [the oldest, but sometimes questioned text]. They are found in the Critical Text published in the Twenty-sixth edition of the Nestle-Aland Greek New Testament (N) and in the United Bible Society's third edition (U), hence the acronym "NU-text."
So, your go to example of corruption in the New King James turns out to be a total dud. Not only are these verses included in the text by the NKJV (and practically every other modern version), but the footnotes concerning those verses state clear and accurate information about the ancient manuscripts relating to those verses.
This is the sort of thing that happens to you, Marke, when you blindly believe whatever you're taught instead of thinking for yourself. Everything you've been told in relation to the notion of the superiority of the King James Bible should be looked at through squinted eyes (i.e. with skepticism). I have yet to find one who wasn't either like you, someone who is just parroting something they've heard someone else say that they thought sounded convincing, or who thought like a conspiracy theorist connecting things that haven't anything to do with one another and using the worst sort of "logic" imaginable to force the conclusion that they want. Reading KJV only people feels exactly like watching an episode of "Ancient Aliens" on the Discovery Channel.
Have you got anything else or was this your best shot?
Clete