That's a good point.
The current presdient of the FIDE was Deputy Prime Minister and close confidant of Dmitri Medvédev during his presidency from 2008 to 2012 when Putin was the Prime Minister because Putin has already served the two consecutive terms allowed by the Constitution so he appointed Medvédev as his puppet President before he was allowed to return to the presidency in 2012 (and reformed the Constitution to be able to stay until 2036, for now).
So yeah, maybe you can probably expect some of Putin's influence in the president of the FIDE.