Why don’t we check for one’s antigen /antibody levels before and after (few weeks or months later) the person was vaccinated for Covid? Would it be useful to provide better correlation for the effectiveness of the vaccine against Covid infection?

It would be useful if you didn’t believe in antibody phrenology. No, having high SARS-COV-2 antibody titers does not mean you are immune; the actual science is a lot more nuanced than that. Which antibody class and subclass in which system and against which specific variant are you measuring? That would be more useful information to actually assess your risk of infection than ‘I have high antibody titers, I must be immune to COVID derp! Different classes and subclass elicit different immune responses. Different classes are more or less prevalent in different systems. WH1 anti-spike antibodies aren’t particularly useful anymore yet they are still the overwhelming booster immune response. The gold standard is anti-nucleocapsid IgA induced mucosal immunity (i.e. upper respiratory tract); you only get that from natural immunity thus far.

How Long Does Natural Immunity Last