Nope. "That" is the wrong indicative pronoun; and you cannot "never" anything (wrong word order). Our poster did not use the title given, but even the official grammar-Nazis (editors) of the source site didn't quite get it right:
18 obsolete words, which never should have gone out of style
Since "never" is a contraction of "not ever"," one would not usually say or write, "... which not ever should have gone ..."
Furthermore, it is not customary to begin a sentence with a numeral. Also, the comma here just takes up space and is unnecessary to transmit the thought. So a better way of putting the headline in print would be:
"Eighteen obsolete words which should never have gone out of style"
There. Fixed that.
No, “that” is the restrictive.
The words that should not have gone out of style are a subclass of the words that have gone out of style.