It is not known if this practice was popularized in New York City restaurants.
25 December 1919, Life, pg. 1072:
The invitation that use to appear in small-restaurant windows:
"All You Can Eat -- Twenty-five Cents."
19 January 1925, New York Times, pg. 4:
"The Tub is one of the cleanest restaurants in New York, where you can get meals for 5 cents -- all you can eat."
27 November 1925, New York Times, pg. 19:
"All You Can Eat for a Nickel."
Urban J. Ledoux -- better known as "Mr. Zero" -- introduced a novelty in the form of a 5-cent turkey dinner. He fed turkey dinners at "The Tub" in the basement of 33 St. Marks Place, on the basis of "All you can eat for a nickel," and said that he was able to break even financially. Dealers furnished the turkey and trimmings at cost, the cooks volunteered, the diners waited on themselves, there was no overhead and and at the end of the day the ledger showed no red ink marks, according to Zero, who claims credit for the world's greatest achievement with the 5-cent piece.
9 March 1931, New York Times, pg. ?:
...one of the tickets now being issued to the unemployed by Zero's Tub, at 33 St. Mark's Place, "Good for All You Can Eat." The tickets were real.
28 February 1932, New York Times, pg. SM11:
"ALL YOU CAN EAT" -- AND WHAT IS CHOSEN
The New Fixed-Price Plan at the Restaurants Reveals That the American's Real Desire Is More Dessert
Down in Memphis, Tenn., at the hotel where possibly first in the United States the fixed-price lunch was inaugurated two years ago, a buffet steam-table on wheels brings even all the pieces de resistance hot to one's elbow, with as many encores as desired.