The first published reference to what we now call bechamel seems to have been in de la Varenne's book, Le Cuisinier Françoise, published in France in 1651. From the English edition, published in 1655, "[M]ake a sauce with very fresh butter, a little vinegar, salt, nutmegg, and the yolk of an egg to thicken the sauce, and have a care that it doe not curd or (turne).
Several people have been credited with the invention of bechamel, de la Varenne among them. If true, we see from the recipe, that both studded onion and roux itself was actually a later addition to the sauce than nutmeg -- and that nutmeg is indeed one of the original ingredients. FWIW, de la Varenne is given a lot of credit for popularizing roux as a thickener. Most cooks of the day used bread. De la Varenne is also given credit for naming the sauce.
The great Escoffier favored the method KTC likes, "White roux moistened with milk, salt, onion stuck with clove, cook for 20 minutes;" while Pellaprat, also a great, added onion not to the milk but to the already cooked, bland bechamel to make a daugher sauce called a Soubaise.
On the other hand, Pellaprat allowed a proportion of stock, as high as 75%, for the liquid. To me that's a veloute, while an egg thickened veloute is an allemande. Was Pellaprat, who wrote a generation after Escoffier a wild-eyed revolutionary or heretic? No. Pellaprat was more traditionalist than Escoffier, in that Careme, writing a hundred years before Escoffied allowed for stock also. In fact his recipe for bechamel started with a veloute: "Reduce the velouté until it is thick, then bind it with egg yolks and thick cream. Stir with a wooden spoon to make sure the sauce does not stick to the pan. Remove it from the heat, add a piece of butter the size of a walnut and a few tablespoons of double (heavy) cream. Add a pinch of grated nutmeg, sieve through muslin (cheesecloth), and keep hot in a bain-marie."
Limping into the second half of the 20th Century, the modern "classic" from LaRousse is the normal roux thickened milk we all make. In his recipe LaRousse advises "season with salt and pepper and, if desired (according to what the sauce is to be used for), a little grated nutmeg." (Gastronomique, 1988). Even in academic, classic French cooking many of these choices are simply a matter of personal style. It's easy to get caught in the idea that what you were taught is the "right" or the "original" way -- a mistake I make frequently.
In my experience cooking Greek (limited) and Italian cuisines (not exactly what you'd call expert, either) bechamel (besamel in Greek) is almost always flavored with nutmeg -- especially when used with layered casseroles such as lasagna, pastisio (Italian), pastitsio (Greek), moussaka, or even the Franco/ Italo/ Jeffersonian macaroni and cheese.
The Italian tradition is of historical interest as to whether nutmeg is one of the original ingredients goes to one of the other bechamel creation myths. That is, bechamel was brought to France by Catherine de Medici when she left Italy to marry the future Henri II in the early 16th Century. According to Italian tradition, the sauce was even older, and created in the 14th Century. Sneer not, Francophiles. It's Catherine's influence and imported courtiers and servants who are credited with lifting French cuisine out of the middle ages and into the light.
Cuisine is living, breathing, and subject to change. We're not constrained to doing things as Escoffier did, nor are we limited by his recipes or vocabulary. The terminology and tradition as to who was first, what's real bechamel and what's merely a variation is confused. IMO it's not for us, mere mortals all, to decide which of the culinary gods was more right than the other. But, in any case, "nutmegg" is neither an American nor a recent offense against whatever the Platonic ideal bechamel truly is.