In tracing the advent of the "chanties," it is important to distinguish the songs used for halyards from those used for capstan. For as a chantey designed for halyards, with modifications, might work for capstan, it rarely goes the other way. The forms are different. Halyards required quite a strict form; most of those chanties are of uniform...er, form. Capstan work could use material from many sources. This is what makes halyard chanties distinctive and potential revealing with respect to their origins. The form of the rowing songs appears to lend itself better to halyards this was the song format that appeared relatively new to the scene, from the perspective of the Euro-American commenters.

Capstan songs, on the other hand, had been around longer, though I cannot say specifically what their form may have been like from one period to the next.

I suggest we maintain a distinction between the songs for capstan and the ones potentially used for halyards, lest we muddle the stream of development of these different kinds of work-songs.

Here is a reference to singing at the WINDLASS from the beginning of the 19th century.
It is found in an edited volume of a poem THE SHIPWRECK by a sailor, William Falconer. This edited edition is from 1806; seems the first edition was probably 1803. It is a footnote reference to the old spoke windlass, in which one must continually remove and replace one's handspike.

As the Windlass is heaved about in a vertical direction, it is evident that the effort of an equal number of men acting upon it will be much more powerful than on the Capstan. It requires, however, some dexterity and address to manage the Handspec, or Lever, to the greatest advantage; and to perform this the Sailors must all rise at once upon the Windlass, and, fixing their bars therein, give a sudden jerk at the same instant; in which movement they are regulated by a sort of Song pronounced by one of the number. The most dexterous managers of the Handspec in heaving at the Windlass, are generally supposed to be the Colliers of Northumberland; and of all European Mariners, the Dutch are certainly the most awkward, and sluggish, in this manoeuvre.

[Was the action anything like cotton-screwing?]

For an illustration of this kind of windlass (not sure just how realistic?) see here, at 2:40:
old fashioned windlass