So, I should preface this by saying that I've largely studied the first and third internationals, but not the second. Also, the perspective i give is extremely US-centric.
Firstly, talks about Kaunsky get into the weeds of how we define "revolution." Take these quote from The Road to Power (emphasis mine, and sorry about the wall of text):
We are revolutionists, and this not simply in the sense that the steam engine is a revolutionist. The social transformation for which we are striving can be attained only through a political revolution, by means of the conquest of political power by the fighting proletariat. The only form of the state in which Socialism can be realized is that of a republic, and a thoroughly democratic republic at that.
The Socialist party is a revolutionary party, but not a revolution-making party. We know that our goal can be attained only through a revolution. We also know that it is just as little in our power to create this revolution as it is in the power of our opponents to prevent it. It is no part of our work to instigate a revolution or to prepare the way for it. And since the revolution cannot be arbitrarily created by us, we cannot say anything whatever about when, under what conditions, or what forms it will come. We know that the class struggle between the bourgeoisie and the proletariat cannot end until the latter is in full possession of the political powers and has used them to introduce the Socialist society. . . But we can have only the vaguest conjectures as to when and how the last decisive blows in the social war will be struck. All this is nothing new
Since we know nothing concerning the decisive battles of the social war, we are manifestly unable to say whether they will be bloody or not, whether physical force will play a decisive part, or whether they will be fought exclusively by means of economic, legislative and moral pressure.
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The democratic-proletarian method of battle may appear more monotonous than the revolutionary period of the bourgeoisie; it is certainly less dramatic and striking, but it calls for far fewer sacrifices. This may be somewhat disappointing to those smart literary persons who come to Socialism as an interesting sport, looking for interesting stuff, but not to those who actually have to do the fighting.
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When we declare that revolutions cannot be made, and when we maintain that it is foolish, and indeed pernicious to incite to revolution, and when we act in accordance with these statements, we do not do this in the interest of the capitalist politicians, but of the fighting proletariat.
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This is largely the Orthodox Marxist line on revolution, and what separated them so sharply from the Anarchists in the First International. The party doesn't create a revolution; the proletarians do. Revolution in Kautsky's mind means the transition from capitalism to socialism as necessitated by the forward march of history, not just any old coup.
His opponent to the right, Bernstein, has largely rejected that notion and supported incremental gains for the proletarians and the eventual evolution from capitalism to socialism.
His opponent on the left, Lenin, had reintroduced a component of blanquism, and supported the immediate creation of a revolution through the actions of an elite vanguard.
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So why did Kaunsky's ideas hold little influence compared to his opponents?
A major reason is that the events of the early 20th century marked a polarization within the socialist movement, and his works were something of a middle ground between those supporting vanguardism or direct action and those supporting democratic means.
The case of the US Socialist Party is a good example. In 1910's, the Socialist Party attempted to gain an electoral presence, but felt it was weighed down by its associations with violent radicalism. It was no secret at the time that their membership was largely made up of syndicalists who advocated for direct action. They had departed from DeLeon's Socialist Labor Party to avoid this very thing, but here they were again.
After a particularly bad bombing committed by two syndicalists in 1910, the leadership felt it would damage their attempt to win a local mayoral race. This situation was only made worse by the radicals' admissions of guilt. In order to try and remedy the situation, the party's leaders amended their constitution in 1912 to expel anyone advocating for direct action. Syndicalist and founder of the IWW, Bill Haywood, would be one of the more prominent members forced out; an event that caused a wave of departures from the party in protest.
Common sense would say that these departures were largely just syndicalists feeling as if they had no place in the party. In a way that's true; syndicalism was largely seen as compatible with both Marxism and Populism, so many did sympathize on that level. The more accurate characterization would be that the left-wing of the party, those who didn't see parlimentariasm as viable, were the ones departing.
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This gets into the second reason why Kaunsky ended up having little influence: the October revolution.
Take the reaction of Earl Browder, leader of the CPUSA from roughly the 30s to the 50s. He was in jail at the time, but the news impacted him profoundly. For the first time in his life, there was a major success in the socialist movement. It seemed that revolution could be made, even in Russia mind you, if a party were to just shed its right-wing elements and dedicate itself to agitating for immediate change.
Upon being let, out Browder immediately entered back into the Socialist Party in order to aid the party's left-wing in pushing for direct action. The strain this placed on the party, along with the radicalism of the language federations, laid the grounds for the departure of the left-wing which formed the Communist Party of America and Communist Labor Party. After losing its left wing, the Socialist Party would move to the right until the great depression legitimized radicalism to a unprecedented degree, and communists expelled by the CPUSA (Lovenstoneites, Trotskyists (Cannon's faction), etc) began to join.
An array of factors would cause the CPUSA to sway left and right between the second period, third period, and popular front eras, but--with the exception of like 5 years under Browder--they always upheld Lenin's vangaurdist ideas.
The Socialist Party attempted to uphold that democratic marxist tradition, but was racked by instability which was only worsened by their mid-30's move towards popular frontism.
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So, I guess the ultimate answer i'm getting at is that Kaunsky's legacy was largely overshadowed by Lenin's for a few reasons:
The Communist Party is undoubtedly seen as "the party of Lenin," the nature of Kaunsky's legacy is such that no one really thinks about the Socialist Party as "the party of Kaunsky."
A collective memory centered around the Cold War leaves little room for left-wing ideologies outside of Leninism.
Parties that could vaguely be considered decedents of Kaunsky's line generally had a hard time competing against Communists, Social Democratic parties, and trade-union-based radicalism
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Browder, Earl. “Notes and excerpts from draft of projected work A Political Autobiography.” Typed, Emory University, no date.
Browder, Earl to Theodore Draper “Correspondence between Browder and Draper.” Typed, Emory University, 1956-1959.
Draper, Theodore. Earl Browder. “Transcripts of Draper interviews with Browder.” Typed, Emory University, June 2, 1953 – October 22, 1954.
Draper, Theodore. Earl Browder. “Transcripts of Draper interview with Browder.” Typed, Emory University, June 15, 1955.
Kautsky, Karl. The Road to Power. 1909. Marxists.org, e-book.
Eley, Geoff, “International Communism in The Heyday of Stalin,” New Left Review Vol.157 January– February, 1986
Klehr, Harvey, and John Earl Haynes. "Revising Revisionism: A New Look at American Communism." Academic Questions 22, no. 4 (Fall 2009).
Palmer, Bryan D. "Rethinking the historiography of United States communism." American Communist History 2, no. 2 (December 2003): 139-173.
Ryan, James Gilbert. "The Making of a Native Marxist: The Early Career of Earl Browder." The Review of Politics 39, no. 3 (1977): 332-62.
Carl A. Linden, “Oppisition and Faction in Communist Party Leaderships” in Faction Politics: Political Parties and Factionalism in Comparative Perspective, Santa Barbara, Calif: ABC-Clio, 1978.
Draper, Theodore. The Roots of American Communism. New York: Viking Press, 1957.
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Klehr, Harvey. The Heyday of American Communism: The Depression Decade. New York: Basic Books, 1984.
Klehr, Harvey. John Earl Haynes, and Fridrikh Igorevich Firsov. The Secret World of American Communism. Annals of Communism. New Haven: Yale University Press, 1995.
Ryan, James Gilbert. Earl Browder: The Failure of American Communism. Tuscaloosa, Ala.: University of Alabama Press, 1977.