Sorry, but that's ridiculous. You can't provide medical advice by simply complying with a legal requirement. If that were the case, workplaces wouldn't be able to carry first aid kits, make food preparation staff wear gloves and hairnets, or even get their employees to wash their hands after going to the bathroom despite being legally required to do so.
FWIW, I'm a lawyer.
An employer can be liable if an employee comes into work sick and exposes people to the illness. Likewise, an employer can be liable if one of its employees is an asymptomatic spreader of a communicable disease like hepatitis, and exposes someone to it.
An employer can be liable if one of its employees is an alcoholic and causes damage while drunk on the job.
OBVIOUSLY, THESE SITUATIONS ARE NOT ANALOGOUS, because the risks of coming into work sick/drunk are far greater than the risks of coming into work without a vaccine. Pro-vaxers will argue otherwise, and I hope courts swiftly shoot that logic down. Just throwing out some examples of how personal health choices can affect people performing their duties at work, since that's what you mentioned.
I'm a law anon, but not an American one. This isn't my area of practice, either - I'm just sharing what I found.
I don't think that the distinction is as salient or relevant as people are making it out to be. The FDA has specified that COMIRNATY and the EUA vaccine are materially identical (i.e. they're only "distinct" in a legal sense), and can be used interchangeably. The vaccines contain identical ingredients and are manufactured using the same process. The products are only "legally distinct" to the extent that they will bear different labels - the FDA press release is pretty clear COMIRNATY is just the existing vaccine, but marketed under a new name, and that both products are approved for use in 16+.
Sources:
https://www.fda.gov/media/144414/download
https://heavy.com/news/comirnaty-vs-pfizer-vaccine-legally-distinct/
Perhaps other scarf patterns represent Islamic faith, but this particular pattern is that of a keffiyeh. I've never seen a scarf in this pattern worn as a symbol of anything other than Palestinian nationalism. I am Orthodox Christian and commonly see these scarves at church. It's unambiguously a Palestinian scarf, worn as a symbol of her Palestinian heritage and not her religious affiliations.
By your logic, wouldn't it be "medical advice" for an employer to force an employee to wear gloves when touching food because wearing gloves supposedly reduces the transmission of diseases like hepatitis? There's research out there suggesting that bare hands can be more sanitary in a lot of contexts, so it's not like it's proven that gloves are always better - it is just a legal requirement. Or, wouldn't it be medical advice for an employer to force their staff to serve food with tongs, even though they aren't legally required to do so, because it's supposedly more sanitary (even though there's research suggesting the opposite - cross contamination risks, etc.). Obviously not - you don't have to be a doctor to try to make your workplace more sanitary, even if your efforts are misguided.
As long as you're not literally pretending to be a doctor/nurse/licensed professional, enforcing broad beliefs or even providing advice about how to prevent or treat illness in general is not "providing medical advice". Beyond pretending to be a doctor to seem more authoritative, illegal medical advice is stuff like actively diagnosing or attempting to treat a specific person's specific illness - stuff literally only doctors can do.