There is a strong correlation between incidence of mercury amalgam and alzheimer's. Obviously correlation is not causation, but I'd still be wary of putting mercury in my body...

No there isn’t a correlation. Maybe you are thinking aluminum? Even then the data is weak.

Mercury salts are corrosive, organomercury compounds are highly neurotoxic and elemental mercury is relatively non-toxic if ingested.

The form of the element matters a lot.

I'd also add that elemental mercury is toxic if inhaled as a vapor. While I mentioned fillings as a safe use, there is a (very small) amount of vapor released when they're installed, and more gets released if they need to be removed. Not enough to matter in adults. If you needed enough fillings for it to be a concern the dental microbiota issues causing the decay would be a far bigger general health threat. Indeed, Alzheimer's is linked to gingivitis (gum disease) and the hypothesized link with amalgam fillings may be an artifact of people with worse gum disease having worse dental health, and thus more fillings.

Gonna need to see a source on that one. I've never seen it and I follow biology news quite a lot.