It’s always tricky when head coaches become too powerful and also assume the GM mantle. Belichick only answers to Robert Kraft right now and that’s awkward. The pair have worked together for over two decades and Kraft doesn’t want to fire Belichick.
An independent general manager would do their job properly. This also relates to the stream of free-agent failures and dodgy trades over the past couple of years. Brady has always had an eye for a player but there have been some strange selections in recent history (via Boston Herald).