How do people here think: "no vax=no job" is a right thing? Or wrong?
I don't know how to answer this question, personally..
For me....
Either side of any contract can apply whatever requirements they like - everything is up for negotiation. Plenty of jobs already require vaccination (eg Hep C for medical staff, TB for pretty much all adults etc).
It's only problematic in terms of altering an existing contract - but that's fine, so long as the change is far enough in the future to cover contract renewals or notice periods.
When France floated the idea, they were talking about 3 months notice - which seemed fine to me. When the UK leaked the idea, they were talking about a few weeks, which was not.
Covid vaccine is different and more controversial, because it's new, and not already standard, and accompanied by a massed false-information campaign.
Nobody has a problem with various other restrictions on freedumb for the sake of pubic safety - just this one - largely because the libertarian and anti-vaxx movements have become organised and politically powerful and well funded over the last decade or two.
Does no vax=no job mean people are forced to have it, that's a grey area.
It's really not.
It's coercion, not force. Absolutely nobody is talking about making it law of the land for all jobs (bad faith slippery slope arguments from the anti-vaxxers aside). Any company can put their own requirements on any contacts offered - including rolling contracts, so long as there's sufficient notice and that it's not discriminating against an oppressed minority (which being a selfish idiot isn't). The closest that's been talked about as law is medical and healthcare workers - who are the most important to be vaccinated, and are supposed to be caring the most for other people's health. Nobody has an automatic right to be a healthcare worker just because they want to, so nobody's rights are being threatened.
As for soft pressure, I think it will depend on the law. As with jobs, refusing someone something on the grounds of not having a vaccination could open companies up to discrimination lawsuits.
Outside of America, they will fail. Discrimination laws are to protect people from oppressed minorities for things that are inherent to who they are (skin colour, gender, sexual orientation, faith, disability etc).
Inside of America legal decisions are often based more on political ideology than legal exactitude.