My concern is not so much that people can only be fired for cause, as opposed to "at will" which is standard for non-union employees. It's the excessive appeals process which can be so protracted that an employer may just give up rather than bothering with it.
There is no way under existing law to address this problem, as these unions can bargain for whatever they want. Some sort of change in the law is necessary, saying the maximum can only be one appeal which must be resolved in x time.
Another concern is with public health issues, like what is happening with the police union in Chicago right now. The mayor there is requiring disclosure of vaccination status for the police, which is a necessary precondition for a vaccine mandate. Nearly half the police force there is refusing to comply, which may cause a suspension of all of them, impacting public safety.
Their union is not saying that these officers object to the mandates per se. They are saying that any such mandate must be bargained for in the CBA. Which is another way of saying, if you want us to comply, you need to first fork something over.
The Chicago Fraternal Order of Police has said the city's vaccine mandate is a change to employment policies that must be negotiated with the union.
chicago.cbslocal.com