Advisory is not IN.
So legally it isn’t in government yet, however as it is officially part of the elected governments plans, you can make a sensible argument that it is part of the incoming transitional government that has been elected and while having now power due to not being sworn in yet, is indeed part of government by nature transition teams and the president elect having status in government by way of things like security briefings and other rights and privileges normally only held by the incumbent government like increased security protection…
It would be like saying a government in exile (a well established precedent of history) isn’t a government and none of the people in it are in government…
His position as co-head of the nominal “department of government efficiency” only exists due to the legitimacy it has been granted by the recognition granted to it by the incoming administration… otherwise we would all be calling it some variety of the first buddy’s pet think tank and arguing over if the incoming administration would even pay attention to it or not… that is a government granted position of power, a position that it is pretty hard to argue is not part of the government that grants it legitimacy… therefore making it a position in government… even if it’s unpaid and advisory… it’s still practically in the government if not legally (for all the conflict of interest reasons you highlight)…
I’m not trying to make a civics or political science case here… I’m talking politics as the exercise of power by government upon the governed… he is currently having breakfast lunch and dinner with the incoming president, making arguments and shaping the cabinet, and contributing to the transition team… he’s involved with government… he’s “in” it.