The immune system has multiple effects channels, only some of which involve antibodies. It takes a while to produce antibodies in detectible quantities and sometimes viral infections are cleared before that happens.

Yea, this doesn't make sense to me. Making antibodies is literally one of the key functions by which the body clears infections.

How can you eradicate that which you cannot detect?

Negative antibody results from this must mean that either the test was a false negative, or the person tested has a dysfunctional immune system which produced little to no antibodies. (How did they survive the infection?!)

There is the innate immune system, the body's first line of defense, which is 'always on', and does not use antibodies.

It is the adaptive immune system that produces antibodies for a specific antigen - I guess for very mild infections the innate immune response is sufficient, and the adaptive never gets triggered.

With other human coronaviruses, it is speculated that we defeat them so easily that we don't develop long-term antibodies, explaining why we get colds with very mild symptoms so often.

At some point presumably there is a carrying cost to all the what-if scenarios. If your body contains/maintains a whole host of organisms going around checking whether an unrecognised something is 'one of these bad things' eventually you're going to run out of spare energy to do other basic tasks.

At what point does this become optimised to "you're probably one of these, therefore I exterminate you"? Note that when that system goes wrong you could end up in auto-immune disease land, or somehow eventually copy informational mistakes/signals land (cancer).

This is my overly-simplistic view though on limited understanding, I am not a medical professional.

We have lots of other antiviral mechanisms that kick in before making antibodies, which is something that we often have to induce our body to do in the context of a vaccine.