I have a theory on why Widmore was kicked off the island.

So Ben said Widmore had a daughter with an outsider. Now we know that Widmore is also the father of Faraday. So he had a family on the island and off the island. Ben Linus found this out and told Eloise Hawking. She had Widmore taken off the island. So that is why she only helps Ben and not her former lover, Widmore.

So just let me know what you think.