Most champions win because there are in a championship car and usually one that is capable winning the constructor championship. Yes there have been occasions when the WDC is won by a team and another wins WCC.
I think Schumacher came back and won a boat load of titles after a gap in his titles. The main difference is he built a team around him to build a championship car and had a clear number 2. Hamilton, Alonso, Vettel etc.. give them a championship winning car and they will win it. As seen this year the driver cant win the championship without a decent car no matter how good some people think they are ( Hamilton in his engine eating Merc)