The word breakfast describes an act, breaking a fast, rather than a menu or a time. That is the first thing that makes its history slippery: the meal moves around the day, changes contents completely, and keeps the same name throughout.
Medieval and early-modern writing on the subject is largely about propriety. Whether one ate in the morning was tangled up with age, occupation, rank and the season, and the arguments were conducted in that vocabulary rather than a nutritional one.
The working day reorganises the meal
What settled breakfast into its modern shape was the timetable. Once large numbers of people left home to work at a fixed hour, the morning meal had to become quick, portable and repeatable. Everything we now think of as breakfast food is downstream of that constraint.
The same pressure explains why breakfast is the meal most thoroughly industrialised. A meal that has to be identical, fast and cheap every weekday is an obvious target for manufacture, and it was manufactured accordingly.
Advertising joins the argument
By the early twentieth century the case for breakfast was being made in print by people with a direct interest in the answer. That does not automatically make the case wrong, but it does mean the volume of the argument stopped being a guide to the weight of the evidence.
Reading the period's advertising alongside its household writing is instructive, because the phrasing crosses over. Lines invented to sell a product turn up a few years later in general advice, with the source forgotten.
Why it stays unresolved
The argument persists partly because the question is badly formed. Breakfast is not one thing, the people eating it are not doing the same work, and the meal sits inside a whole day that the question ignores.
As a piece of history, though, it is unusually clear. The shape of the morning meal has followed the shape of the working day almost exactly, and the nutritional argument has generally arrived afterwards to explain the arrangement already in place.