An Elephind search finds two interesting early occurrences of "X time[s] around" that suggest practical, real-world meanings for the phrase.
First, from "General Post Office," in the [Vincennes, Indiana] Western Sun (August 2, 1817):
If you divide the post roads of the United States into two distinct post routs, the mail will travel each week, in stages, nearly equal to three time[s] around the globe ; and divide all the post roads in the United States into four equal or distinct post routes, on which" the mail is carried in stages, sulkies, and on horseback, it will be equal to a travel of six times, each week, around the globe, averaging one post office for every fifteen miles of post road.
Here, the expression used for "X times around" is a cardinal number rather than ordinal number, but the thing being traveled around X times is the globe; and it would not have been a stretch for the writer, if he had been interested in focusing on the last of the three or six times around the globe, to have used the wording "the third time around the globe" or "the sixth time, this week, around the globe."
Second, from "Evangelical Catechism," quoting John Mines, Leesburg, Virginia, October 1820, printed in the Richmond [Virginia] Family Visitor (August 3, 1822):
New Method of Instructing
If there are teachers enough, the number [of students] in a class should not exceed eight or ten. The teacher, either in a family or Sabbath School, should always set as foreman one of good capacity, and change him for some other, more or less frequently, according to the number of such under his care. Thus prepared, let the teach read the question and answer distinctly, before the whole class. Then let him propose the question to the first [student], and assist him to repeat the answer, all the others giving attention. Then let him propose the same question to the second, and help him as before; and so proceed through the class. If necessary, let the same question pass a second and a third time around, in the same manner, until all can repeat the answer.
It seems clear from the context, that each catechism question is passing a second and third time around the classroom, after having initially made its way past each student in the room a first time around. So if the author had expressed himself fully, he would have said, explicitly "let the same question pass a second and a third time around the room."
In the 1850s, newspapers used "X time around" in the context of horse racing, with the implicit meaning "X time around the track or race course." From "Riding and Driving Match by Ladies," in the the [Richmond, Virginia] Daily Dispatch (October 10, 1855):
An exciting trotting match came off, in which Miss Demarest and Mrs. Whitney drove, each lady availing herself of every advantage that offered. The first heat the horses continued neck and neck, but on the second and third time around Miss Demarest came in ahead.—The driving of Mrs. P. Kinney, Miss Stephens and Mrs. S. Miller was also excellent, and those ladies also received a portion of the applause.
In a similar vein is this instance involving boat racing from "Regatta of the Hoboken Model Yacht Club," in the New York Clipper (September 27, 1856):
Th race now lay between the Walton and Restless, one passing ahead, and then the other, until after turning the stake boat at Amos street, when the Walton was ahead by some lengths, beating the Restless ; but after a second time around the yacht at Hoboken, at the starting point, the Restless gained on the Walton, and was, to all appearances, the victor, but at that moment some rigging gave way, and she lost the race, the Walton winning by some few minutes, and taking the first prize ; the Restless, second, and the Arma, of the second class, third, beating the Audubon, of the first class. The other boats were at this time not "around."
A few months later, the same newspaper uses the expression "the first time around these parts" as an idiomatic expression meaning "the first time in this vicinity." From "Blossoms and Buds in Boston," in the New York Clipper (January 31, 1857):
The Firemen Around Here
A four-wheeled hose carriage is to be introduced for the first time around these parts. It is to be placed in the house now occupied by Maverick Engine Co. No. 9, in Paris street, East Boston, and No. 9's machine will be removed to the basement of the school house in Summer street, E[ast] B[oston].
From the proliferation of examples in the 1850s, I suspect that "X time around" became popular from its use in racing as a short form of "X time around the course." But at the 1822 example indicates, "X time around" could also refer to something passing around a room. Idiomatic use of "X time around" seems to follow naturally from these early literal senses of "X time around."