If that was the case the doctors would see that the vaxed getting admited would have a higher death rate than the unvaxed as they would only admit those most likely to die from that group. If they did have a higher death rate then the stats and the doctors experience would reflect that and so the doctors would not heavily select for the unvaxxed. Though there is the possibility in your hypothetical that even the vaxxed with worse symptoms on average than the admitted unvaxed have a better chance of survival than a less symptomatic (when admitted) unvaxxed but if that's the case it pretty much shows the doctor are behaving correctly in a 'bias' of admitting unvaxxed.