TL;DR "System administrators can spot a headless request with ease": NO. They can surely spot some headless requests with ease. But in general, that's not true.
Quite obviously, it all depends on how the requests are done (how much effort to disguise them) and how the logs are checked and what countermeasures adopted (how much effort goes into penetrating the disguise).
The fact is that an automated (headless or not) browser and a human can, and often do, perform requests in different ways, with different patterns. When they do, this difference can be spotted more or less easily.
The difference is often "intentional", i.e., it derives from a design choice - typically not executing some scripts or not downloading (some) images to reduce traffic, seeing and following transparent links or hidden parts of the DOM. Navigation patterns are clearly influenced by a human's response times. Also, patience and scope of the navigation might be inhuman by need - if you need to scrape a whole website, you won't use a "human like" navigation even if you could. And if you see someone browsing sequentially the whole site, you might not care for it even if you know it's a human.
But should a browser (headless or not) be controlled by a mouse and keyboard-like interface by an AI designed to interpret a screen rendering through OCR and scripting, and scrape a site like a human would, it would be almost impossible - and almost certainly intolerably rife with false positives - to tell it apart from a real human being.