I have not seen data on how many of the unvaccinated people currently in the hospital had either confirmed or suspected cases of covid in the past. I think an underlying question here is whether natural immunity is as robust as immunity from a vaccine. It could be as robust in some cases, and not in others. I would add to that to ask whether natural immunity following an infection with one variant will be as protective against another variant as immunity from vaccine x, y, or z. In that regard I did read a peer reviewed paper that suggested that the RNA based vaccines were more likely to lead to protection from variants than natural immunity because the spike protein produced by the RNA will adopt a greater variety of conformations (shapes), allowing the immune system to create a larger variety of antibodies against the spike protein. Since the spike protein is the main target for the immune system, having all the other proteins produced in a natural infection won't necessarily be any advantage.
There is some data contrary from Israel but as you say there’s not a whole lot on natural immunity yet
One anecdote: outdoor bbq. Double vaxxed person wearing n95 except to eat. 9 attendees. 3 had confirmed natural immunity and a recent antibody count of 10 on the scale but not vaxxed. Other 5 were double vaxxed. Double vaxxed attendee had covid (had sniffles at the party…thought it was allergy…found out because she’s a nut about testing every 2 weeks). 3 of the double vaxxed/outdoor/masked attendees fell ill with covid confirmed with test. Of the 3 the 2 younger were not serious the older was. Patient zero sheltered with the naturally immune individuals. Said individuals did not fall ill and took 2 spaces out covid tests to confirm.