Why did Shakespeare include the sub-plot of Antonio and Sebastian scheming against King Alonso?
He wanted to tell us how Antonio is so cruel and wicked in nature that he wanted to kill the king Alonso so that Sebastian would become king of Naples.
Shakespere wanted to demostrate to the audience that Antonio is very power-hungry and foolish. This is represented in the part of the story when they planned to kill the sleeping Alonso at night.
Shakespeare made use of parallelism in his play. Here, he included this sub-plot for us to better understand the wicked nature of Antonio and Sebastian and also, to remind us of how they schemed against Prospero.
(I believe that the answer has come too late.)