As a native Chinese, I can say that the whole arm is also referred as 手. Kote 小手 specifically refers to that part of the arm - the wrist.
And 手臂 is "arm" -- but the question, I thought, was etymology
-- not current use. It is a example of a metonymy -- using the name of one thing as indicative of another thing because of a close association between them. The issue is which way the metonymy has occurred from "arm" to mean "hand" or from "hand" to mean "arm". Clearly, it is the latter.
The classical denotative meaning of 手 is "hand". 手腕 means literally "手" "bends" [ 腕 ] which is the "wrist" -- not the elbow or shoulder, which also bend, creating a ambiguity if 手 was originally primary for "arm" and only secondary for "hand." As mentioned above, 手臂 "arm" -- if 臂 is decomposed -- is deemed, etymologically, to refer to the 肉 "flesh" that 辟 "governs" 手 "hand." The "flesh" of the arm drives the hand, not vice versa.