The short answer is, etymologically speaking, yes. In modern Japanese, however, -ます stem + なさい can be thought of simply as an imperative form, less blunt/direct than the 座れ/食べろ form but still an imperative and thus not as polite as request forms like ~てください.
As in your example, you'll often hear it used by parents and teachers when giving firm instructions/orders to children.
As a general study tip, there's nothing wrong with learning etymology to the extent that it helps you put the pieces of the language together, but you should always keep in mind that the linguistic origins of a phrase or word don't necessary have much (or any) bearing on how it's used in the modern language. (As a fun example, 貴様 started out as an honorific form for "you", as the kanji would suggest, and now it's essentially an expletive with a nuance along the lines of "you bastard".)
With ~なさい, at least, you can see some set phrases where it retains a polite/honorific nuance, like お休みなさい, お帰りなさい, and so forth.