I always wonder what products are so important that they need to be delivered next day? The few times I realised I need something urgently it typically means today and even next day delivery is not enough, so I go to a store and possibly pay a slightly higher price. For everything else it doesn't really matter if it is one or 5 days. Is the next day delivery really necessary or does it just appeal to our impulse of needing to have it in our hands right now?
For many people, going to a store the same day is completely impossible (not having a store within reasonable distance that stocks any similar product).
This is not just rural areas, either: even in major cities items can be delivered faster than the time it would take to find a store that carried the item, go to the store, get it, and come home again. It's really only suburban areas with a high density of big-box retail that even have the option to "go out and get something" immediately.
Amusingly, it's also only in these suburban areas where next day delivery even exists to compete. As an anecdote, living in the downtown of a major American city (population > 1million), the average Amazon delivery time is roughly 3 days, and last mile deliveries are all delegated to the USPS which adds significant time.
For me it's more about the certainty than the rush. I hardly ever need anything next day, but "next day" is a concrete day, while for example "in 3-5 days" isn't. As someone who is often not at home, I'd rather not have to be at home or arrange for someone to be there several consecutive days in case a package arrives.