[posterboy]

It's rather obvious that the archaic meaning has been largely lost and therefore that folk etymology informed the morpheme's convergence to "oak". The GP is still silly though, because e.g. the "fly thing" is an inappropriate loan translation - "flight gear" would be more appropriate (cp. https://en.wiktionary.org/wiki/Zeug), whereas a "plane (sheet) of air" isn't any more meaningful.

[adimitrov]

Absolutely. Folk etymology is actually a productive process; in the literature, you'll also read "reanalysis", as in, a word's components which are not understood are reanalysed as something familiar. Compare English "sparrowgrass" — asparagus, or bridegroom, where groom is originally "gome", meaning "man" in Old English, and not "groom" in the modern (i.e. Early-Modern-English) sense.

The translation of Zeug, and especially -zeug is contentious. Stuff, gear, tool, utility, means. There are many ways in which it is used in German: Flugzeug, something that flies (or that you use to fly with); but Schwimmzeug is not a boat. Instead it means the "stuff" you need to go swimming, like goggles and a swimsuit. Schlagzeug (Schlag: beat) is neither something you use to hit somebody with, nor all the things you need in order to go a-hitting. It's a drum kit.

[picturesal]

Not "plane"=sheet, but "plane" from the Greek "planos"=wandering. (Same root as planet=wandering star).

An aeroplane is a wanderer through the air.

[tullianus]

Etymonline disagrees; -plane here is from French planer "to soar," ultimately from Latin planus, "flat." http://www.etymonline.com/index.php?term=aeroplane