My sister who is a public health specialist, cautioned me about how difficult establishing cause-and-effect in these sorts of discussions can be in general. She hasn't talked to me about this specific case, but lemme bring something up from the perspective that we Computer Scientists can understand.

Cause-and-effect chains form a DAG (Directed Acyclic Graph). For example, A causes B causes C, in theory (or in this case, A causes B prevents C. Cause vs prevent doesn't really matter, its just a negative on the correlation diagrams)

In this case, we want to determine if "Flu Vaccination causes/prevents Alzheimer's Disease".

Now lets consider a bunch of real life factors. Wealth, Age, Sex, Race, Educational Attainment, Insurance. Just the basic stuff anyone would think about. Lets focus on Educational Attainment + Insurance to keep things simple.

* Education -> Flu + Education -> Alzheimer's? Or is it the case that "stronger mind -> Education -> Flu Vaccines" and "stronger mind -> Education -> Alzheimer's" ?? Education is almost certainly a conflating situation (Education would probably cause higher flu vaccine uptake, and simultaneously would reduce Alzheimers), so maybe the Flu vaccine isn't causing the effect... but is instead just a cross-correlation to education level.

* Insurance probably causes flu-vaccines (because flu vaccines are free for most insurance plans). Insurance probably doesn't cause/prevent Alzheimer's though. So we have "Insurance -> Flu-vaccine -> Alzheimer", which means we don't have to worry about a cross-correlation in this case.

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Etc. etc.

Basically: to truly understand if we have a cause-and-effect situation here, we need to think of all possible conflating factors (such as education), while discounting factors that are "along" the DAG-chain towards the hypothetical result (ex: Insurance).

After that, there's apparently a lot of math and analysis that we can do. If we take the records of the people who participated in this study, and get their educational levels + insurance information, we can then account for these conflating factors.

I probably messed up the language and everything else for that matter. But just a reminder that public health discussions are rather technical. There's both a science and an art to the whole process, and just reading the "results in the raw" could lead to a misunderstanding.

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The best example of this is "Ice Cream causes Drowning". (Real life: Hot summer temperatures cause ice-cream consumption. Hot summer temperatures cause people to swim.) We need to account for the Hot-temperature effect if we want to find the truth behind that correlation.

There's also the chance that we've got it all backwards. Alzheimer's causes people to stop taking the flu vaccine. (IE: Alzheimer's might cause people to be more easily swayed by anti-vax propaganda, causing them to take less flu vaccines)

Figuring out all possibilities (as well as possibilities that aren't even listed) is the job of a public health care specialist. They'll get it wrong, but hopefully less wrong than you or I would. There's apparently mathematical techniques to be used to tackle each of these questions I brought up.