Your right to emigrate is not meaningfully impaired, for the same reason that it isn't meaningfully impaired by the requirement that you wear shoes at the airport.
The claim was this was a burden on par with shoes. I claim it is far beyond that and many of the US mandates will be struck down. The precedent is a state level mandate with a $150 penalty.
Organizations/companies can still impose basically any requirements they want to in the US. In general, state/federal requirements are basically air cover.
Here's the full CDC brief[1]. The short version: your religious belief generally has to be "sincerely held" (which means that you don't get to pick and choose which vaccines you take based on political winds). Some states have even stronger constraints, including you to present an affidavit and demonstrate that your religious beliefs are not merely an extension of some misguided personal belief.