Sure it can. McKinley escapes assassination in 1901. The Republicans never initiate progressive policy, so the progressives both grow more radical and more disillusioned with two party politics, ultimately turning to the Socialists as do Northern industrial workers and ex-Progressives. Essentially a milder form of RaRT resulting in a party shift rather than full on revolution.
Kevin Phillips's biography of McKinley depicts him as a surprisingly progressive president who was likely to propose important new initiatives with respect to the tariff and trusts. McKinley's last speech to the Pan-American Exposition in September 1901 heralded a campaign for tariff reciprocity. Actions against trusts would likely follow in 1902 when the United States Industrial Commission appointed by McKinley in 1898 reported back. This report, as Phillips notes "wound up laying out much of what would be the Progressive corporate and antitrust agenda through 1914." (Phillips,
William McKinley, p. 136)
Also, at least according to Mark Hanna, McKinley himself might have undertaken a prosecution against Northern Securities as TR famously did in OTL: "I warned Hill that McKinley might have to act against his damn company last year. Mr. Roosevelt's done it. I'm sorry for Hill, but just what do you gentlemen think I can do?"
https://books.google.com/books?id=B0Jr8Ypal1UC&pg=PA392
Furthermore, Phillips notes McKinley's pro-labor record, which included naming Terence V. Powderly, onetime leader of the Knights of Labor, as commissioner general of immigration, and of Samuel Gompers of the AFL to the Industrial Commission. McKinley frequently consulted with Gompers.
Anyway, even if we assume that McKinley is a reactionary--or is followed by one as the GOP candidate in 1904--it hardly follows that voters are going to desert the major parties en masse--or even if they did that it would be for the Socialist Party. The Democrats in 1904 might not flirt with conservatism as they did in OTL if they were facing a more conservative candidate than TR (in OTL they actually hoped for business support against TR--in vain of course [1]). And even if they did, they would learn it didn't pay and revert to Bryanism in 1908 as they did in OTL.
And finally even if we assume that somehow both major parties for some reason remain unaffected by reform sentiment and keep nominating conservatives, a non-socialist third party led by someone like La Follette seems more likely to take advantage of it than Debs' Socialists.
[1] As the
New York Sun, considered the voice of Wall Street, stated in explaining its reluctant endorsement of TR over the "safe" Parker, "We prefer the impulsive candidate of the party of conservatism to the conservative candidate of the party which the business interests regard as permanently and dangerously impulsive."
https://books.google.com/books?id=X43uHzjM_GIC&pg=PA82