Antibodies are proteins too. Why does the human body produce them? Ask the people that have deficiencies in antibody production! They have serious problems with infections, especially bacterial infections. So antibodies are important. It's true that HIV usually finds a way to escape from any antibodies that an infected person manages to produce. It's only because HIV is a very tough virus and can find a way to escape from almost anything, including all the drugs people have designed in the last 25 years.
I really don't understand the biology of all this. I'm not even sure what antibodies are and why the human body would produce them if they are no use.
Proteins don't die because they aren't alive. Proteins can denature, I guess that's what you mean. If the sample is stored properly, p24 will not get denatured for a long time and could be detected by tests.