Do you think we should add an FDA phase five stage, where exorcists and witch doctors carefully study Covid vaccines for evidence of black magic and evil spirit? Because it's perfectly sensible to let rational people die to satisfy the whims of whoever holds the most irrational beliefs? Because you're making the exact same argument, except just substituting out Western anti-vaxxers for folk superstitions.
There's no evidence to indicate that anti-vaxxers are sensitive to clinical evidence whatsoever. A third of American adults are considering foregoing the vaccine[2], despite it being the most extensive and sophisticated safety trial in vaccine history.
Why would you believe that the minutiae of the FDA approval process would have any effect on anti-vaxxers, when the vast majority of their beliefs come from wholesale fake news.[3]
[1]https://www.econstor.eu/bitstream/10419/172046/1/884014355.p... [2]https://apnews.com/article/ap-norc-poll-3rd-adult-skeptical-... [3]https://www.thelancet.com/journals/landig/article/PIIS2589-7...
This stop in Germany comes from Germany's federal vaccine authority, which as far as I know, does not employ witch doctors.
If it turns out that a vaccine actually has harmful side effects (which is possible, this is not some conspiracy but a risk of all drugs), and the administration would have ignored it, you have eroded trust far beyond anti-vaxers.
People aren't utilitarian machines, trust in drugs is very easily broken, vaccines in particular. Showing neglect and lack of precaution may harm vaccination efforts for decades.