I think it depends. As someone who was instructed that firing an individual for failing to comply with state and company guidelines last winter.
I know you are dealing with much higher wage employees who likely take it very seriously... more than a youngster just bouncing around ski bumming after college... but if you fire people for violating company policy, and you make those guidelines company policy, I think legally you have a good shot. You could make them wear red shirts on Tuesday if you want to. Uniform policy. Non disclosure policies.
I think legally the issue would be if a COVID test is a violation of personal information. As long as there is another option besides getting the jab (no matter how ridiculous it is), employers will be ok here. It's annoying but it's the same as if a company was like hey, you need to ride public transportation or you need to get your car inspected weekly.
My biggest question is... does the employer pay for the weekly COVID testing? If they do and it's company policy, it's just like any other policy IMO that an employee has to follow. It will be interesting to follow legally. People may start becoming more likely to litigate their employers the longer it drags on.