What was the first instance of the use of a closed time loop (An event where the cause and effect of a time loop feed into one another, creating a stable and unending chain of causality), in a piece of published American science fiction literature, and what was the response to it of professional literary critics who publish through a regular periodical such as a newspaper or magazine?
Here is a possible answer to your title question, copied from my answer to the scifi.stackexchange question What was the first 'time loop' story featuring the bootstrap paradox?. Unfortunately, I'm not aware of any professional literary criticism of this piece of pulp science fiction.
1935: "The Man Who Met Himself", a short story by Ralph Milne Farley (pseudonym of Roger Sherman Hoar); first published in the August, 1935 issue of Top-Notch magazine; reprinted in Farley's 1950 time-travel collection The Omnibus of Time. This story was part of my answer to the question What was the first Time-Travel story to involve gambling?.
Review at the Internet Time Travel Database:
Among physicists, the most favored resolution to time-travel paradoxes is a world of one fixed landscape of time and its events. Time travel may be possible, but if so, the Karma will conspire to have only those events that have been written into the landscape to occur. Heinlein’s “—All You Zombies—” may be the pinnacle of such stories, but Farley’s is the earliest case that I’ve read to present a clear deterministic time loop along these lines. In the story, Boston stock broker Dick Withrick is on a 1935 tiger hunt in Cambodia when he runs into a strangely familiar (and slightly older) man who warns him, “As you value your freedom, do not touch the machine—” And yet, he does touch the machine, taking him back to 1925 so he (in the company of his Buddhist Abbot host) can relive the decade of financial turmoil.
The time loop is described in this excerpt::
Not a single flaw in his time-traveling. Not a single puzzling or preposterous thing about it, except for one question: Whence came the time-machine which he had found in the Cambodian jungle?
He put that question to the parchment-faced old Abbot.
Yama Toga laughed. "Perfectly simple," he asserted. "It is now locked up, oiled, and in good condition, in some hiding-place of mine, unrevealed to you; is it not?"
"I suppose so," Withrick admitted, "although I haven't seen it since I left it in the woods the day I back-tracked in time."
"1935 will arrive," the Abbot continued, an inscrutable smile playing about his Oriental features. "A young American named Richard Withrick will come up the trail, bound on a tiger-hunt. I shall have the time-machine taken back to the proper place, several days in advance. One of your beaters will discover it. Dono Dal will explain to this other you how to operate it."
"But how will Dono Dal know? An ignorant lesser priest like he!" Withrick interjected.
"Ah! You have already told me his exact words, many times. I have written them out on palm-leaves, and have drilled and drilled the stupid Dono Dal in them, until he is letter-perfect. When the proper hour decreed by fate arrives, he will recite his little piece. The cycle will be complete."
"But what if he makes a mistake?"
"He will not make a mistake. What is written, is written."
"But you haven't yet told me where the time-machine came from!" Withrick exclaimed, exasperated.
"You yourself brought it here out of 1935.'
"But where did it come from originally?" Withrick persisted.
"There never was any 'originally'," the yellow-robed old Buddhist patiently explained. "You flew the time-machine backward through time from 1935 to 1925. And that is why I now have the machine all available, to plant so that it will be found by you in 1935."
"To fly it back again to 1925?" Withrick added, interrogatively.
"No. Nothing of the sort. There is no round-and-round circle of events; no repetition. Merely one closed cycle. One overlapping of events for only ten years. One Richard Withrick, with ten years of his life folded together in the middle. One time-machine, found in 1935 and brought back to 1925 — found in 1935 because brought back to 1925. That is all."
"But who made it in the first place? — Oh, skip the 'in the first place'. Just plain: who made it?"
"No one. It was never made. There never lived, and never will live a man who could make a time-machine. Time-traveling is impossible. You yourself have accomplished this impossible feat, because you found, in 1935, the time-machine which you yourself brought into 1925 out of the future. And when 1935 arrives, and you have found the machine, and have flown back with it, the machine will thereby be gone forever, gone back into the past, and no time-machine will exist. It never existed before 1925 — it will not exist beyond 1935. And yet it was never created, and will never be destroyed."
"Just like Einstein's idea of the universe: finite but unbounded," Withrick gasped. "There must be some flaw in it somewhere, but I'm damned if I can figure out what it is. The only time-machine the world has ever known, or will ever know, existing for a brief space of ten years! Existing in complete circularity; existing in 1925 merely because it existed in 1935, and then was flown back ten years in time; existing in 1935, merely because carefully preserved by you for ten years."
"Well, after all, is not that the same solution as is given to the reason why anything exists? It is here because it is here."
Withrick chuckled. "We used to have a song about that, back in America," he said.
And this is the nearest that he ever came toward solving the riddle of the time-machine.