Sorry, but no, it isn't the "final evidence." While it is true it there is a species of barley found in the New World--called "little barley"--it did NOT come from the Old World, but is native to the New World, most likely originating in South America.
5. Barley in the Americas
Our final physical evidence is also small—as tiny as a single grain. In fact, it is a grain. In Mosiah 7:22 we read that the Nephites cultivated barley, even utilizing it for monetary purposes: “And behold, we at this time do pay tribute to the king of the Lamanites, to the amount of one half of our corn, and our barley, and even all our grain of every kind,”
Barley comes up in the Book of Mormon four different times (Mosiah 7:22; 9:9; Alma 11:7, 15). Yet prior to A.D. 1492, this grain wasn’t known to have existed, let alone cultivated, in this hemisphere. Thus, more fodder for Book of Mormon criticism and ridicule.
It wasn’t until 1983 that archeologists acknowledged the existence and cultivation of a type of New World barley that dated to as early as 800 B.C.11
Hordeum pusillum - Wikipedia
Note the article says that this barley is not closely related to old world barley, hordeum vulgaris spontaneum, which is native to the Fertile Crescent area of the Middle East. IF ancient Israelites had come from there to the New World as per the BoM, then THAT is the type of barley they would have brought with them, which has a much bigger grain, and it would have been here when white men first came to this continent, but it wasn't.