This is not always true. There are (desired) emergent phenomena that are robust to a lot of minor tweaks to the underlying elements. For instance, neural nets are effective, even independent of the details of the individual neurons.
I would have said that you care whether a phenomenon is emergent because studying individual elements carefully is usually a bad way to understand emergent processes. We understand macroscopic hydrodynamics much better than we understand the precise forces between water molecules.
This is not always true. There are (desired) emergent phenomena that are robust to a lot of minor tweaks to the underlying elements. For instance, neural nets are effective, even independent of the details of the individual neurons.
I would have said that you care whether a phenomenon is emergent because studying individual elements carefully is usually a bad way to understand emergent processes. We understand macroscopic hydrodynamics much better than we understand the precise forces between water molecules.