Hauling a bowline was a "short haul" task, comparable to "boarding a tack" (tack being the line attached to the a clew of the course, below where a bowline is attached, to the leech).

Sailing ships nowadays generally don't have bowlines. I suppose there must be a few that do, perhaps replicas of 18th and early 19th century vessels? It's not something I've paid close attention to, but my non-authoritative impression is that they generally are not part of the rigging.

There is a classic book about rigging from 1915 (?); I don't remember the name. But it makes the point that in Elizabethan times bowlines were more "important" lines and, I think, heavier. I believe this is the point that Hugill was repeating when he tried to reason an age for the "Haul the Bowline" song. I consider Hugill's point to be moot to the argument about the age of "Haul the Bowline," however, since bowlines evidently were still in use in the mid-19th century, i.e. at the same time when "Haul the Bowline" appears in literature.

I'm no expert on the timeline of bowline use. However, Harlow (who sailed in 1875) talks about bowlines. He says that sing-outs were sung when "hauling aft the sheet of the courses, boarding the main tack, or hauling out the bowline" (pg 5).

In _The Sailor's Word-Book_ by Smyth, 1867, the author refers to the use of bowlines, and does not say it was old/obsolete. Further, he defines the term "Bowline haul", pg 124:

//
A hearty and simultaneous bowse. (See One! Two!! Three !!!) In hauling the bowline it is customary for the leading man to veer, and then haul, three times in succession, singing out one, two, three—at the last the weight of all the men is thrown.
//

Elsewhere he denies "One, two three!" (pg 506)
//
The song with which the seamen bowse out the bowlines; the last haul being completed by belay O!
//

In Falconer's _Universal Dictionary of the Marine_, 1760s, he defined "un, deus, troi":
//
...an exclamation, or song, used by seamen when hauling the bowlines, the greatest effort being made at the last word. English sailors, in the same manner, call out on this occasion,—haul-in—haul-two—haul-belay!
//

It may be less significant that Smyth uses the phrase "singing out" and more significant that Falconer calls this simple 1,2,3 "an exclamation, or song."

There are some other references in French to vocalizing while giving stiff pulls on a bowline.

While the idea of "chanties" not being sung in navy vessels is basically true, I wouldn't read that to exclude *vocalizing* during the short hauls, i.e. bowline, tack, course sheet, and various difficult "sweats" that required multiple people.