The word is neither archaic nor dubious.
Leeneia, my wife used to card and spin her own yarn, using a spindle as described in the song. The simplest device, which is called a drop spindle, or just spindle, has a 'spindle shaft,' and a 'whorl' at the bottom, which has weight and can be twirled as the yarn is spun onto the shaft. Most spinners and books on the subject call the 'whorl' a 'weight,' but some call it a 'rock.' In more mechanized systems, the 'whorl' is on a pully.

In the song, the "rock" applies to the entire spindle, shaft and weight. the "reel" takes and winds the thread from the spindle shaft. Usually the distance between the pegs on the wheel of the reel, and an entire turn, are set distances so that it is easy to tell how much thread is put on the reel. Most have a clicker that sounds with each full turn, as an aid in counting.

Now how are you going to make this understandable to an audience that knows nothing about spinning?? "Whorl" would mean no more than "rock." Add a new verse of explanation?

"Why not change to the verses in other 19th-20th c. versions:
I sold my flax, I sold my wheel,
To buy my love a sword of steel,---
or:
I'll sell my frock, I'll sell my wheel---
Both of these would be understandable. I would guess that 'rock' is a mistake for 'frock.'

"Gone the dove---": poetic and evocative. American versions of "Johnny" are all "modern (19c and later)," so I fail to see the problem. Or use the Peter P & M version with the memorable(?) "come bibble in the boo."