Has it ever happened that a team got to the world cup through the playoff and then won the cup?
Or any team who did the playoff is historically doomed? I couldn't find any statistics, maybe you know this stuff better.
There was no such World Cup where the winner would qualify through play-offs. It's also worth to mention that qualification rules were constantly changing and sometimes winners played only few matches (if played at all, as they were hosts or defending champions till 2006 World Cup qualifiers) in order to qualify.
During the qualification phase of the 1934 World Cup Italy played only a single match in order to qualify, which can be considered as a play-off match, but in fact was a group stage match.