> What exactly makes the GAO not "part of the executive branch" ?

https://www.law.cornell.edu/uscode/text/31/702

>> (a)The Government Accountability Office is an instrumentality of the United States Government independent of the executive departments.

The law establishing it also establishes it as independent.

Sure, that is what is written down. But as a necessary part of its operation, there is a whole lot of executive power being exercised as well, which the unitary executive theory says would fall under the authority of the president.