The issue isn't whether the employer can require you to vaccinate. The situation posited is if you must work on site the employer says you must take the vaccine, or if you are remote, the employer wants people back in the office and requires them to come back vaccinated or lose their job. The issue isn't the mandate....the issue is the liability. Because the vaccine is still emergency use authorized, insurance won't cover any damages. I'm not aware of Congress or the state of California adopting a liability shield for employers (they did for the pharma company and administers) and given its EU and the employer themselves is not requiring the vaccine to be physically admined on site workers comp may not cover it. So the issue to date with employers not requiring it is that because of the EU authorization its not considered part of regular protective medical care, and by forcing employees to take it, the employer could be found liable (and not covered by insurance or workers comp) if the employee suffers any side effects for injury.
What's changed is that in the Biden news conference he signaled that the EU labels might now start getting removed for at least one vaccine in the next couple weeks (probably Pfizer since they are beasts at reg approvals) which clears the way not only for a federal mandate for its employees but also makes it easier for employers (since in many states the claims might then shift to coverage for workers comp).
BTW, when the EU label gets removed for 12-18, expect school mandates too, both public and private, for that category.
Does a little yellow caution light come on in your brain when you start making things up?