OK I read the article and the linked study. The article claims that the vaccine doesn't protect one from infection. The study doesn't say that. Actually, IIRC only 22% of the people in the study were vaccinated for the flu.
BUT we are often told that even if the vaccine isn't highly effective in total protection (which I think we would all agree means you won't get it at all) it helps minimize the effects if you do get the flu so it is worth getting.
From that, I can believe that vaccinated people who get the flu can spread more virus than unvaccinated people simply because the vaccinated ones don't feel too ill and are out and about where the unvaccinated ones are (hopefully!!) home in bed, probably spending too much time on or in front of the toilet.
If the CDC funded the study, and the results are as suggested by the ExplainLife website (which I can not find any info about other than on the site itself) I wonder where/when the CDC is going to put out statements relating to the results and if they are going to change their stance on people getting vaccinated if for no reason other than to minimize the impact of the disease should one get it. It would seem that a vaccinated person shipping out over 6x the virus as unvaccinated people should get some alarm bells ringing at the CDC.