Hawkins said there are exceptions to this rule, such as Ebos chief executive Mark Waller.
But he added that Ebos was a strong performer at the time of Waller's transition.
"If Sky was trucking along well, our view might be different," Hawkins said.
"But it's struggling. It would be better to make a clean break."
Remaining board members include the controversial Derek Handley, who was recently reappointed and chairman Peter Macourt, who has flagged his intention to resign once a replacement is lined up.
Stewart said in a statement, "After a period of transition, during which I have been grateful for John's advice and support in his capacity as a board member, John has decided to step down from the Sky Board.
"John will continue to be available to me and the leadership team to support us. We are grateful for his outstanding service to Sky, and he will always be remembered for his passion and commitment to this business."
Fellet had a 27-year management career with Sky, spending his first decade as COO then a decade as CEO, helping to transform it from a company that was losing $1m a week to one that was consistently profitable and paid fat dividends before running into recent headwinds including the rise of Netflix, live-sports piracy and the Spark Sport insurgency.