Right. No one is pushing government vaccine mandates. Even in the midst of the most acute public health emergency in a century, it’s understood Americans won’t abide that, especially the ones who are actually inclined to not take it. Ergo, a mandate would do nothing but stigmatize the vaccine further among the vaccine-skeptical.
But as you point out, private employers and businesses are free to set their own guidelines on vaccines just as they did before with face masks and distancing (some places set bare minimums, but nothing prevented them from going above and beyond what their local regulations required).