Interesting article
here on the history of saves, with a specific section on the 3-inning save, and when it went into decline:
There was never a time where the three-inning save was a common occurrence, but it happened far more frequently in the 1970s and 1980s than it does today. Notably, Rollie Fingers, who saved 341 games in his major league career, pitched three or more innings in 36 of those saves; 18 of those 36 were saves solely based on the innings rule — not because he protected a tight lead. Meanwhile, Goose Gossage went three-plus innings in 23 of his 310 career saves, and he earned 11 of those saves because of innings pitched (which would put him at 299 if the rule were ever amended).
However, in recent times, they have become ultra-rare. In 1977, there were 180 of them in the 26 team league, according to SB Nation. By 2010, there were only four in the entire season.
One reason for this decline is because of Tony La Russa’s use of Hall of Fame closer Dennis Eckersley with the Oakland Athletics in 1988, according to the Washington Post. Of the 60 times Eckersley pitched that season, he entered only one game in which his team trailed. Only 11 of his 60 appearances were 2.0 innings or more, although none lasted three innings or more.
The strategy worked out well for the Athletics, who would make it to the World Series that season. Although the A’s fell to the Los Angeles Dodgers in the fall classic, Eckersley was the runner up for the American League Cy Young Award and finished fifth in the MVP voting — something which did not go unnoticed by other teams across the majors.
From 1988 to 1990, the number of three-inning saves went from 110 to 50, as SB Nation points out.