The US constitution isn't concerned with common sense, only legality. There are very many stupid laws on the books. Private firms are allowed to have rules requiring vaccines because they are, well, private. The constitution does not prevent private firms from requiring them. The main protection the constitution is concerned with is government overreach. After all, it's the government that has the guns and jails.
What happens in Greece and in Europe is beside the point. America has a long tradition of more individual freedoms. The president isn't a king. Of course, Biden is trying to get around that by having OSHA (Occupational Safety and Health Act) division institute a new safety requirement - getting the vaccine. The courts will determine if it passes constitutional muster.
Ford is allowing workers not to get the vaccine probably because too many won't get it and they might have to shut down their production line. They can ill afford less production. As it is there are no new cars to buy, certainly few below MSRP sticker list prices.
My own personal belief is I wish everyone would get the vaccine who's medically fit to get it. But I'm not sure I'm in favor of mandates. That's a slippery slope.