Considering he had the large plates, why didn’t King Benjamin already have the small plates instead of Amaleki having to give them to him.
In Jacob 1:1-3 we learn that Nephi gave Jacob charge of the small plates, which were for prophecies and similar writings. From Jacob 1:18-19, and really the entire book, we can deduce that Jacob was also left in charge of the church, or was the spiritual leader of the people after Nephi’s death. At the end of Jacob and in the books of Enos, Jarom, and Omni, we learn that the small plates were passed down through Jacob’s line.
In Jacob 1:9 we learn that a king was anointed to replace Nephi – and that king was not Jacob. So, with Nephi’s death, the political and spiritual leadership of the people separated. While it isn’t said directly in Jacob 1:9, it’s implied that the kings kept the larger plates that were for history. Jacob 3:13 confirms that the history was kept there. Then Jarom 1:14 makes it pretty clear that the kings controlled the larger plates:
And I, Jarom, do not write more, for the plates are small. But behold, my brethren, ye can go to the other plates of Nephi; for behold, upon them the records of our wars are engraven, according to the writings of the kings, or those which they caused to be written.
Omni 1:11 confirms that the record of the people is on plates “had by the kings”.
Finally, in Omni 1:25 we learn that because he had no descendants, and because he respected King Benjamin, Amaleki chose to give the small plates to the king. At this point, the political and spiritual leadership of the people joined back up in a single person, as it was with Nephi.
I hope this helps you understand the two “paths” these sets of plates traveled, and why. I also encourage you to study The Words of Mormon, where Mormon talks about the two sets of plates, and how (verse 7) he joined the small plates to his record for “a wise purpose” (something which Nephi also wrote – see the footnotes). We now know that this “wise purpose” was so that they could become the start of the Book of Mormon after the loss of the 116 pages (see D&C 3, D&C 10, and portions of “The Contributions of Martin Harris” from Revelations in Context).