WI: Labour Party instead of Progressive

Let’s say that Roosevelt in the aftermath of the McKinley Assassination but before he died (he stayed more than a week in the hospital before his death) and John Hay takes the office, a more conservative politician than McKinley and strongly anti-labour-union, harming the relations with the progressive camp of the party.
What would be the effect of the progressives politicians of both party join to found a Labour Party (inspired by UK Labour Party) in perhaps 1907? How would this affect the American party system?
 
My thoughts it has to do with the split of the labor movement between conservatives who endorsed capitalism and radicals who aligned with socialism. With the decline of the Knights of Labor, labor split with conservative labor aligning with the Democrats by the 1900s and radical labor aligning with the Socialist Party. That being said, labor unions had been a major backer of the People's Party faded into irrelevance after fusion with the Democrats in 1896.
 
The US had a labor party, the Socialist Party. Both Democrats and Republicans were scared of it and progressivism was arguably a response to it. So it is unlikely that they would jump on board with the Socialists unless the Socialists gained power first or became less antagonistic. (However, the party did endorse La Follette in 1924.) So if you want the Socialist Party to become a major party, you need to have larger unions and more moderate leadership willing to cooperate with other parties.

I think the most likely result of your scenario is a bigger Progressive Party which eclipses the Republicans in the north, so that they compete with the Democrats in a two-party system.
 
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