Sorry - I should have clarified this. People have "natural antibodies", antibodies that are not developed in response to Covid-19, which consist of IgM, IgG and I think IgA. These are not specific for particular viruses, but can bind to and eliminate them. People also have Natural Killer cells, a class of lymphocyte that destroys infected cells. These are the mechanisms by which people recover from infection within a week or two, before covid-soecific antibodies develop.
Spud - I think you need to do a crash course in immunology.
IgG, IgM, IgA etc are so-called isotypes (or classes) of antibody. All are specific, in that they have an antigen binding site that will recognise a particular antigen on a foreign body, e.g. a virus protein. They are all specific for particular viruses, or rather a protein component on a virus. The exception being if the same protein is found on more than one virus in which case antibodies raised to virus A may protect from virus B.
So on the case of covid, the immune response involves the generation of specific antibodies, of all of the above classes, to various covid virus proteins. Without that acquired immune response our ability to fight infection is very weak, just associated with the innate immune system. The latter does involve natural killer cells, but their ability to respond to infection is limited without the acquired immune system.
And contrary to your post the acquired immune system (even in a first infection) will be kicking out Covid-specific antibodies within days, so your person who recovers from COVID in a week or two will be using the acquired immune system as the key component to fight that infection.