To be fair, management has just as much to gain by disseminating false information. (That everything is hunky-dory.)
Also, workers trying to organize is not 'infiltration by unions'. It's their lawful right, in a society which allows free association. Management bargains collectively - why can't employees?
More likely, employees will unionize, and come to some agreement with management, or, they will fail to, and anyone attempting to unionize will be fired. (Illegally, but in at-will states, employees have almost zero recourse for that kind of termination.)
Actually, private-sector employees have recourse for that in all states under the federal National Labor Relations Act -- they can file a charge with the NLRB. The NLRB staff are very helpful and understand that most people are not lawyers, so they will even help you draft the charge form and walk you through the process.
What kind of rhetoric is that? Is it inherently bad for workers to unionize and speak with one strong voice instead of hundreds of weak voices?
It could equally be disseminated by the Big Three (who have done stuff like this before) or by shorters (likewise).
Or it could be all true.