The departing chief executive, former rear admiral Chris Parry, stepped down with immediate effect amid media speculation that he had alienated members by launching a series of combative attacks on the state sector and the Charity Commission.
Parry said public benefit rules were "a missile aimed from the maintained sector into the independent sector", and his stance towards the state sector was described by the Department for Children, Schools and Families as "frankly insulting".
The ISC said that Parry stepped down "by mutual agreement" in order to "take up other opportunities that he has been offered".
Burgess is a lawyer who has accused the Charity Commission of misunderstanding public benefit principles and of holding views "that are not supported by the law".
However, he also worked closely with the commission when it was drafting public benefit guidance on fee-charging charities.