After this period, the English were quite enamoured with the French for a little while there and so the language then borrowed a lot of French words (see: enamoured), and colourised a bunch of others with French spelling (see: colour/color).

I'm not an expert on the history of English, but this doesn't seem linguistically or historically accurate. More than half of modern English words are derived from Latin or French, even if the absolute most common words tend to have Germanic roots. Old English before the Norman conquest of 1066 is unrecognizable to native speakers of Modern English (except through academic study), and even Middle English (post-Norman conquest through about the popularization of the printing press in the 15th century) is really hard for native English speakers to parse. Middle English eliminated a lot of grammatical rules or Old English, to better match French grammar, because the ruling class in those days still primarily spoke French.

The French influence on the English language began in 1066 and never let up. By the time American English split from English English, they were already a heavily French-influenced language, before Englishman Samuel Johnson began standardizing spellings, and the American Noah Webster moving to simplify spellings, primarily by choosing the simplest version already in common use. Webster's work was popularized mainly in the U.S., so that England and the U.S. standardized spellings differently, and some portion of words remain spelled slightly differently.

So no, I don't think the English started incorporating French words or other linguistic influence only in the 1600s.