The vaccines are no longer experimental. They have "passed" and are now approved for general use. Being approved under emergency legislation does not mean safety assessments were skipped or minimized. Providers also have the ethical responsibility of "justice," which means here to ensure public health in this public health crisis. Receiving a vaccine (and being mandated to given no medical contraindications, even though it is a hard line), is essential for patient safety as a provider can be infected without knowing it, masks are not perfect, and hospitalized patients are more likely to suffer and die if they get COVID while in the hospital. People are not casually admitted to a hospital; they are there because they cannot get adequate care as an outpatient. Providers and those in a hospital are ethically mandated to do what is needed to keep those patients who have placed their trust in them safe. As an analogy, it would be unethical for a parent who knows of the dangers of secondhand smoke on infants to smoke right next to them, blowing smoke in their face. While the parent may be fine and the infant may be fine, it increases the risk of sudden infant death syndrome in this vulnerable infant population. "Freedom" is ultimately bound by ethical obligations and legislation. While hunters have the freedom to hunt and provide for their family, they are responsible for stray bullets that could harm/kill others in the general vicinity. No one would argue that "freedom" equates with a right to murder people, demonstrating limitations in using it as a catch-all argument.
While we could assign liability to individuals who infect others, this is extremely hard to determine for an infectious disease and likely could not be defined in court. In the hunting example above, it would be straight forward on figuring out who shot the bullet. Given that people do not become infectious for days after becoming infected, liability could not be effectively assigned in many cases.