Because the gap indicates unfairness and people care about that. If you are poor, raising the floor will be good enough, you primary worry will be sufficient income. Once you reach a comfortable life, your priority will - or at least may - shift, now you are in a position that allows you to worry if the situation is fair.

Also narrowing the gap is a way to raise the floor including in absence of growth while raising the floor without narrowing the gap depends on growth. But I think this is not too relevant in practice because taking a lot of money from a few wealthy people will often result in rather small changes as the money has to be spread among a lot of people.

Fairness is ideology. The world will never ever be fair, nature isn't fair. We should strive for fairness of course (raising the min/medium).

But that still isn't a convincing reason (I want to be convinced)

Because the gap is the percentage of society's resources that are controlled by wealthy stupid people. Think about all the power companies that are cutting off power to homes to power AI data centers. That couldn't happen if the DC owners had less wealth and the homeowners had more.

Because needs and preferences of lower level are less and less reflected in policies and production. They are becomming permanent underclass with no future. They dont matter for politics nor for economics. Preferences of super rich are overwriting them, while super rich become richer, while super rich offload all risk they take on others.

That still doesn't explain why the gap needs to narrow if the min/median are rising. The pie can increase, everyone can be better off without shrinking the gap.