What are Everstones useful for? Why would someone not want their pokemon to evolve?

Sometimes you want a pokemon to stay in a specific form until it learns a certain move, one that the evolved form can't learn. The Everstone just makes it so you don't have to keep cancelling the evolution each time the pokemon levels up. One example would be Eevee who learns a number of moves that are unavailable to any of its evolved forms. As it levels towards these, it may well max out its friendship and try to evolve into Espeon or Umbreon.

Secondly, the Everstone can be used to pass Natures down from parent to offspring.

Primary reason is to learn moves faster. Some people don't want to evolve pokemon. Nowadays the everstone would be used for breeding natures.

they're convenient if you're training a pokemon past it's minimum evolution level to get a move faster or because the evolved form doesn't learn it. saves you from getting the cutscene and hitting b every time it wants to evolve.
everstones are also useful when breeding to get certain natures for the babies. it has changed since gen 3, but currently, whatever parent holds the everstone will pass down its nature 100% of the time.
if the pokemon has a regional variant, you can also give the everstone to ensure it's the right one. for example, if you breed kantonian and alolan sandslash in alola, they'll always have alolan shandshrew babies. if you give the everstone to the kantonian one, then they'll always have kantonian sandshrew, even in alola.
normally the form depends on where you are, alolan varients every time in alola without an everstone, or kantonian varients everywhere else without an everstone

Some people would rather have a pikachu than a raichu. or something like that.

Maybe to keep it small/cute. Most everstoners (lel) like their pokemon's design and don't want it to change. It's all about preference.