That's what I always understood----that the body can create the antibodies quickly once you've had it to prevent further infection.
For most people anyway.
What people often forget is that vaccines require a person to have at least some level of functioning immune system in order to develop resistance. However at a certain point that wears out for the extreme elderly or those with other illnesses. Thus, most people in nursing homes eventually die of pneumonia or even flu even though there are vaccines for both of those illnesses. The vaccine gooses your immune system as long as you have one or the remant of one.
That is a roundabout way of saying that we should still expect some level of deaths from covid in the extreme elderly even if they have had covid or have had a vaccine. But with infectious disease control we should not have New York type clown shows. No excuse for that.