well tested

Moderna is not yet approved by the FDA, nor is JJ. I can only surmise the FDA has not finished testing these vaccines.

safe

The FDA has not approved the vaccines - thus they are not known to be safe.

effective

The initial emergency approval required 50% effectiveness. Pfizer seems to drop below this effectiveness after 6 months.

why we’re at the point of mandating vaccines in the first place.

If we have an approved safe effective vaccine, then we might consider mandating it. Either the FDA is being slow, which I suppose is possible, or we do not have one yet.

The government has a legitimate, compelling interest to protect the public health.

Vaccines only protect the public health, as opposed to protecting the individual, if there is a reasonable expectation of reaching herd immunity. If R0 for delta is 6, then a vaccine would need to be at least 80% effective to have a hope of herd immunity. We do not have a vaccine with that effectiveness right now, so herd immunity is impossible with current vaccines. Thus everyone is going to get COVID eventually, and there is no public health benefit to vaccines, though they do reduce the individual risk of death by perhaps a factor of 15.

Is there a public interest in reducing health care utilization? Perhaps, but that is much less compelling, and the government does not ban soft drinks or cigarettes for this reason.