Why not both?
There's plenty of evidence that throughout the history of the US the military has had an outsized influenced on US international policy. A lot of this is a hangover from WWII and the Cold War.
The thing is that nowadays none of this is done in the shadows. Lobbying from defense contractors and foreign governments are out there for all to see. So you can have:
- an elected official that is even mildly corrupt and is influenced by lobbyists
- an elected official that is dogmatic and truly believes in American Exceptionalism and it's necessary spread throughout the world
- an elected official that is more interested in domestic policy and lets the bureaucrats and advisors largely determine foreign policy
- an elected official who is well-meaning but succumbs to public and media pressure to ensure the country is safe or at least feels safe
All of these can (and likely would) lead to a foreign policy that seems like it's run by those in the shadows.