You can have a labor movement without a union. In the U.S., any meaningful labor movement will be non-unionized. In the U.S. unions are used to hold onto the scraps you have, not to bargain for more.

I am reminded of Bukharin's take on industrial democracy and Comrade Lenin's repudiation of it. From "Disagreements on Principle" section of "Once Again on The Trade Unions, the Current Situation and the Mistakes of Trotsky and Buhkarin".

"Take this famous “industrial democracy”, which Comrade Bukharin hastened to insert in the Central Committee’s resolution of December 7. It would, of course, be ridiculous to quibble about this ill-conceived brainchild (“tricky flourishes”), if it merely occurred in an article or speech. But, after all, it was Trotsky and Bukharin who put themselves into the ridiculous position by insisting in their theses on this very term, which is the one feature that distinguishes their “platforms” from Rudzutak’s theses adopted by the trade unions.

The term is theoretically wrong. In the final analysis, every kind of democracy, as political superstructure in general (which must exist until classes have been abolished and a classless society established), serves production and is ultimately determined by the relations of production in a given society. It is, therefore, meaningless to single out “industrial democracy”, for this leads to confusion, and the result is a dummy. That is the first point.

The second is that if you look at Bukharin’s own explanation given in the resolution of the C.C. Plenary Meeting on December 7, which he drafted, you will find that he says: “Accordingly, the methods of workers’ democracy must be those of industrial democracy, which means. . . .” Note the “which means”! The fact is that Bukharin opens his appeal to the masses with such an outlandish term that he must give a gloss on it. This, I think, is undemocratic from the democratic standpoint. You must write for the masses without using terms that require a glossary. This is bad from the “production” standpoint because time is wasted in explaining unnecessary terms. “Which means,” he says, “that nomination and seconding of candidates, elections, etc., must proceed with an eye not only to their political staunchness, but also business efficiency, administrative experience, leadership, and proved concern for the working people’s material and spiritual interests.”

The reasoning there is obviously artificial and incorrect. For one thing, democracy is more than “nomination and seconding of candidates, elections, etc.” Then, again, not all elections should be held with an eye to political staunchness and business efficiency. Comrade Trotsky notwithstanding, an organisation of many millions must have a certain percentage of canvassers and bureaucrats (we shall not be able to make do without good bureaucrats for many years to come). But we do not speak of “canvassing” or “bureaucratic” democracy.

The third point is that it is wrong to consider only the elected, the organisers, the administrators, etc. After all, they constitute a minority of outstanding men. It is the mass, the rank and file that we must consider. Rudzutak has it in simpler, more intelligible and theoretically more correct terms (thesis 6):

“. . . it must be brought home to each participant in production that his production tasks are appropriate and important; that each must not only take a hand in fulfilling his assignments, but also play an intelligent part in correcting any technical and organisational defects in the sphere of production.

The fourth point is that “industrial democracy” is a term that lends itself to misinterpretation. It may be read as a repudiation of dictatorship and individual authority. It may be read as a suspension of ordinary democracy or a pretext for evading it. Both readings are harmful, and cannot be avoided without long special commentaries."

https://www.marxists.org/archive/lenin/works/1921/jan/25.htm

While certainly social conditions have changed, I fail to see see how unions in their present form, even the IWW (I'm a wobbly) can effectively reach or even be functionally apart of the same system as a proletarian vanguard other than being ultimately subsumed into a committee or labor board under a worker's party guiding economic direction. Most unions, while beneficial to most workers, retain reactionary ideals and sabotage their own interests because of their lack of theoretical doctrine or understanding of class politics. AFL is a good example of this.

"..and cannot be avoided without long special commentaries" is of particular humor to me, he knew how brutal this topic is with other leftists.