To me a chainsaw is a safe tool to use. Dangerous, but safe. To others it is not. Is a chainsaw a "safe" tool?
And the FDA itself talks about approved drugs being safe and effective: https://www.fda.gov/drugs/drug-information-consumers/fdas-dr...
The FDA is incredibly strict when it comes to advertising. I remember reading about one violation where the ad had a tagline for an ADHD medication "So you can concentrate on the important things" and showed a picture of a child studying. The FDA said "the image implies that your medication will improve the ability of a child to study and you have submitted no data to the FDA to support this claim".
Happens extremely rarely though.
No drug is safe, but in this case, not taking a drug is not safe. If it's safe enough to vaccinate 15,000 people, when 300 of them are expected to die without vaccination, it's incredibly unlikely that side effects discovered on 1 billion people will not make the vaccine the safe choice.
Having worked in the pharmaceutical industry, if I ever said my FDA approved drug was "safe and effective" I'd have the FDA dropping the hammer on me.
The FDA does not prove a drug is safe. No drug is 100% safe. The FDA determines whether the risks of the treatment are outweighed by the benefit.
I have no doubt that additional safety signals will pop up from Covid vaccines as the treated population expands by 100x. Most likely they will not be severe and the risk will still outweight the benefit.
But to say "we've tested the vaccine on a 15,000 people so we know it's safe to vaccinate 1B+ people" is not something the FDA would agree with.