Here are the guidelines from the CDC where Covid positive hospital staff, even with symptoms, are allowed to work in certain staffing situations.
CDC contingency and crisis staffing guidelines
Our local health care system put out its own policies developed from these guidelines, though I am not going to publish them on the internet. There are multiple news stories that are easy to find from early to mid January that confirm this was a practice allowed in multiple states.
Now that that is out of the way, I would like to suggest that I have not seen any data that young, healthy unvaccinated people are the major cause of hospital systems being overwhelmed. Those being hospitalized are still mostly unvaccinated, but they have multiple risk factors, especially the elderly, obese, hypertensive, and diabetic. Those people are where the efforts at vaccination should have been made, not at young, healthy college students.
The vaccines were originally approved because of their effect on severe disease. But they were also found to be incredibly effective at preventing transmission of the original variant. Because that was so incredible, many promises were made about the vaccine ending the pandemic. In one way of thinking, the vaccines actually did end the pandemic of the original variant. But along came evolution.
Subsequent variants began chipping away at the vaccine's protective layer of shellac. The delta variant began causing "break through" infections, though the vaccines still protected against severe disease. Then along came omicron, and now omicron BA.2, and significant protection against transmission is practically gone.
Because vaccines currently provide little significant durable protection against transmission of omicron, and because young, otherwise healthy people are not the ones clogging up the hospitals when they get Covid, the vaccine mandates on young, otherwise healthy people will not accomplish the goals that they were originally set to have.