why do we need importance sampling?

A rather common question about using importance sampling, posted on X validated: why is importance sampling helping in the event the function used in the expectation has restricted support, i.e., is equal to zero with positive probability? Which is a recommendation I make each time I teach about importance sampling, namely that estimating zero is rarely necessary! In my Saturday Night answer, I tried to give some intuition about the gain brought by a correct support for the importance function, carried in the ideal case when the truncated importance function remains available with its normalising constant. But it is unclear this set of explanations managed to reach the OP.