Handkerchief looks like it it is a kerchief for the hand and that is indeed the origin of the first element hand-, no false etymology there.
What of the second element kerchief? If you are like me, you have probably come across it occasionally in reading and just assumed it was an old word for handkerchief. But when we dig into its etymology, we find it was originally nothing to do with hygiene, or decorative clothing accessories, or waving goodbye, etc., etc. It meant a square of fabric worn on a woman's head, as in this portrait of Lady Alisa Normanska, a Serbian (?) noblewoman who became a French, specifically Burgundian, countess (image Public Domain, found on Wikimedia, see Comment for reference)
It came into English from Anglo-Norman courchief, 'cover-head', and the two parts of this word can be traced back to Latin and in the case of chief, further, to the Proto-Indo-European (PIE) language ancestral to most modern European languages. Here is an early 15c quote I found in the Middle English Compendium under kerchief.
A large coverechief of threde She wrapped all aboute hir heede.
The first element cour is related to Modern French couvrir and as you will have guessed, cognate with English cover.
The second element is from Latin caput, which seems to come directly from PIE *kaput-, 'head', which is the origin of many words e.g. cap, chef, capital, cape, chapter, capitulate, capo, cattle¶, cabbage, and also in German languages head and Haupt, by the same consonant change c -> h we see in e.g. Latin canis, 'dog' -> English hound, German Hund. So part of our word handkerchief stems from a root 5,000 or years old.
From the 14c we also have neckerchief, and the pleonastic headkerchief, so the fact of -chief meaning 'head' was already being forgotten quite early on.
I'll end with a related joke
Why can you wear an upturned canoe on your head?
Because it's capsized!
¶ Cattle originally just meant goods, it's a doublet of chattel, as in 'goods and chattels'.