Hi, political scientist here.

The United States is rarely an exception on this, but you have to look at the region rather than the culture to find parallels: while labour parties emerged after industrialisation in the United Kingdom, Australia, New Zealand, and in lesser measure, in Canada and even South Africa, the United States party systems is better understood when comparing it to its' Latin American counterparts, particularly Colombia and Uruguay.

The key concept here is labour incorporation, the process by which the political systems open themselves to workers acting like citizens, voting and being voted into office. Ruth Berins Collier and her husband, David Collier, described the different ways on which this process might develop in their 1991 book, Shaping the Political Arena.

When industrialisation and demographic concentration allow workers to find common ground and coordinate themselves, they start creating organisations to advocate for their rights, sometimes through mutual aid, sometimes through protests and strikes, which governments and business interests tended to resist and repress. In addition, in many countries, workers lacked voting franchise, either by not counting with properties required by law (which is known as censitary vote), or simply by lacking enough education to be authorised to vote, so they weren't able to recur to electoral politics still.

In this environment, the Colliers argue that 4 different routes can be taken, either by the country's political elites or by the organised workers themselves: the State can directly allow workers to organise and demand them to use legal routes for it, as it happened in Brazil and Chile; one of the existing traditional political parties can bring the workers' organisations under their wing in exchange for votes, as in Uruguay and Colombia; the workers' organisations or some outsider force allied with them can build a new, populist workers' party, fused with those workers' organisations, as in Perú and Argentina; or the workers' movement can ally with a farmers' movement to completely displace the old political regime, generally through a revolutionary process, and create a new workers' and farmers' party, on which their organisations are fused, as in México and Venezuela.

As you might already notice, the route that the United States took is roughly similar to the one seen in Colombia and Uruguay: during the growing social upheaval of the Gilded Era, the Democratic Party appealed to organised labour and immigrant workers on the industrial north of the country, and some of their biggest organisations, like the AFL and the CIO, got firmly allied to the party on the New Deal coalition. Some small labour parties that emerged on the period, like the Minnesota Farmer-Labor Party, also ended becoming part of the Democratic Party, reinforcing the alignement on their influence areas.

To directly answer your question: the lack of a workers' party on the United States is not proof of the absence or weakness of the workers' organisations, but rather the effect of a successful strategy by the Democratic Party into expanding to become a substitute for such a party.

Following the development of the Colombian and Uruguayan party systems is illustrative of what might happen in the future: in both countries, the success of the parties that managed to ally with organised labour was enduring for several decades, and even today, they are the left-most party of their cohort. But differently to the United States, in Uruguay on the 1990s and in Colombia on the 2010s, new leftist parties have become electorally successful, in such a measure that most organised labour has displaced into allying with them, or directly participating on their creation and growth.

The Uruguayan Frente Amplio, a tightly-knit coalition of leftist parties, emerged on the 1970s but was repressed by the dictatorship; even if before it they managed to command about 20% of the vote in 1971, they had to wait until 2004 to reach the presidency and a legislative majority.

The Colombian electoral leftist parties evolved later, due to the powerful influence of the communist guerrilla movement in the country, that both absorbed popular support from left-leaning citizens, and made the left 'toxic' to more moderate voters. Most of the development of the current Colombian leftist parties falls under the 20-year rule, but suffice to say that many of the most important elements of their coalition were already active in electoral politics during and in the aftermath of the 1991 Constitutional Assembly: the FARC-EP fraction of the FARC guerrilla demobilised in 1985 and became the Patriotic Union party, the April 19th Movement guerrilla, on which the just elected Colombian president, Gustavo Petro, fought, demobilised on that occasion and became the Democratic Alliance M-19 party, then part of the Independent Democratic Pole, then part of the Alternative Democratic Pole.

Sources:

Collier, R. B., & Collier, D.(2002). Shaping the political arena: Critical junctures, the labor movement, and regime dynamics in Latin America. Collier, Ruth Berins, and David Collier.

Henao, N. R. C. (1990). El proceso constituyente: El caso colombiano. Revista de la Facultad de Derecho y Ciencias Políticas, (91), 24-38.

Andersen, K. (1979). The creation of a Democratic majority, 1928-1936. University of Chicago Press.