The way that study's described it seems that it showed us that people who would choose to get vaccinated against flu are less likely to suffer from Alzheimer later on. So, maybe recommending flu vaccinations would cause the same effect, or maybe it's caused by something upstream of the vaccination that wouldn't be affected by recommending flu vaccinations alone.
Sad thing about controlling for confounders is that it's not the case that adding more controlled confounders makes the result more likely to be correct: sometimes one can _introduce_ a confounder that way. Consider e.g. a hypothetical case where people who are more diligent will both be vaccinated against flu *and* will be more likely to have their ailments discovered. In that case correcting for the confounder of "ailments that have been discovered" actually creates a fake correlation between flu vaccinations and lack of ailments.