I think this is probably wrong, but ultimately it doesn't even matter because the pharmaceutical companies have gotten around marketing regulations by not marketing their drugs. Instead they just have ads for "get vaccinated!" and since they're not marketing a specific product, they don't have to list any potential side effects of the product.

Similarly, the government and other organizations advertise "get vaccinated!" for them and also don't need to provide any side effect warnings, because they're advertising a concept, not a specific product, even though we all know which product(s) they're talking about.