The Lenin quote given at the start is taken entirely out of context. It looks like he just searched for a Lenin quote that would fit his flawed analysis.

Here are a few more quotes from the very chapter Rainer takes his quote from. Note that these are quotes from '"Left-wing" Communism, An Infantile Disorder' which was written after the revolution in 1920.

About the necessity of the struggle against the reactionary leadership of trade unions:

The Mensheviks of the West have acquired a much firmer footing in the trade unions; there the craft-union, narrow-minded, selfish, case-hardened, covetous, and petty-bourgeois “labour aristocracy”, imperialist-minded, and imperialist-corrupted, has developed into a much stronger section than in our country. That is incontestable. The struggle against the Gomperses, and against the Jouhaux, Hendersons, Merrheims, Legiens and Co. in Western Europe is much more difficult than the struggle against our Mensheviks, who are an absolutely homogeneous social and political type. This struggle must be waged ruthlessly, and it must unfailingly be brought—as we brought it—to a point when all the incorrigible leaders of opportunism and social-chauvinism are completely discredited and driven out of the trade unions. Political power cannot be captured (and the attempt to capture it should not be made) until the struggle has reached a certain stage. This “certain stage” will be different in different countries and in different circumstances; it can be correctly gauged only by thoughtful, experienced and knowledgeable political leaders of the proletariat in each particular country.

About working where the masses are:

If you want to help the “masses” and win the sympathy and support of the “masses”, you should not fear difficulties, or pinpricks, chicanery, insults and persecution from the “leaders” (who, being opportunists and social-chauvinists, are in most cases directly or indirectly connected with the bourgeoisie and the police), but must absolutely work wherever the masses are to be found.

Rainer's idea of allying with the movements he proposes is flawed in two parts. First, he doesn't call for the struggle against the incorrect and harmful ideas of those movements. He points out some flaws in some of the socialist parties in the US which do exist but his analysis here seems to favor the reactionary parties over said socialist parties. To me it sounds like he's just trying to send "the masses" to these reactionary movements without considering the level of organization, class consciousness and political involvement of "the masses". The situation is not the same as at the start of the 20th century or at the end of World War 1. The working class then was a lot more organized and various communist and reformist parties existed with actual mass support. Today, especially in the West, this is not really the case.

Second, he doesn't really talk about reaching the masses. He talks of allying and supporting these reactionary movements which he just assumes have some mass support already and he assumes that the support they have is from working class people. Neither of these can be taken for granted.

The movements he wants to support are not actual reformist movements, they don't propose any significant reforms to the system, they barely even propose particular policies that they want to adopt. Reformist movements, while flawed, have historically had genuine mass support. The support of the working class that had at least some consciousness and wanted to systematically improve it's conditions. These types of movement can be useful for communists but again, a proper strategy needs to be made. Blind support doesn't lead anywhere.

Movements like RAWM also aren't working class movements. They fall into a group of astroturfed movements that are supported by various right-wing elements. I don't know much about Cornel West specifically, but from what I saw since he announced his campaign, he isn't really supported by the working classes of the US and I don't think he's really connected with them either. He's a (relatively) privileged life-long academic and is now attempting to approach "the masses" from above. "The masses" didn't choose him so they won't be radicalized when his attempts at "reform" fail. As far as I can tell, he also doesn't really have any reforms in mind, just calls to empty phrases like "truth", "justice", and some calls for policies like "a living wage" which aren't elaborated on at all.

Two very timely short threads by Roderic Day on this "purity fetish" phenomenon:
Thread 1
Thread 2