Quick add-on: Test specificity is how good it is at a positive result being a true positive result (not false positive). Sensitivity is the same for negative results (true negative, not false negative). A test with good specificity and/or sensitivity can be useful as a tool. Worse sensitivity and specificity make it less useful.

I think you’ve hit on some key points with mutations, and but it also is useful to know the gritty mechanics. An antigen is something the body encounters that triggers an immune response, typically but not always a foreign substance or organism. Your body produces antibodies until it finds one that works against the antigen it has encountered, then “saved” that antibody in case it’s useful in the future. Thus, the tetanus vaccine protects against future tetanus infection.

The hard part is finding an antigen the immune system will see and that doesn’t change too quickly over time in our population. Flu changes but slow enough for new vaccines to come out each year (with varying efficacy). Some infections have very clever ways of hiding their antigens from detection, such as disguising as your own cells. Others have potential antigens that are accessible but change very quickly, while other parts of their structure are possible antigens that change slowly over time and make better vaccine targets. The better the infection is at hiding or changing quickly, the harder it is to nail down a good antigen to target.