Not true - all western countries require vaccines to go through the rigorous three-stage testing process.
It is due to the fact that the FDA wants two types of application from the vaccine manufacturers. First the emergency use one, and then the Biologics License Application. A time consuming process.
I feel this could be expedited if the FDA applications were not so onerous. One fully detailed and supported application should be enough, as it is in other countries. The health agencies do their independent reviews of the data and quality of the products, regardless.
The fact that the vaccines are helping in markedly reducing the deaths clearly shows that they are working. And if an employer wants to protect their staff and customers (which is what the initial comments you posted were about) that should be their right to do so. As their own OH&S policy requirement. imo
It is not like a person can just say "oh I am not going to wear safety boots even though they are required at this job, because that is my right" ... "oh I am not going to wear gloves when I handle the food the restaurant will serve, because that is my right".