I think most people don't need a separate eye cream, however eye creams do have their place. Couple examples just out of the top of my head. It could be you need a heavy face cream because of dry skin, but applying that around eye area causes milia and it's easier to get two creams that work good on their separate areas then to buy dozens trying to find one that works alright on both. If you are contact lens wearer or/and have sensitive eyes it can be you can't use most face creams or sunscreens around eyes and you can also have allergies or eczema just on eyelids/eye area which means they need more targeted care. It's an extra step that's not necessary for most.
I can't apply anything around eyes currently due to bad reactions and have had easily irritated eyes even before that so I've been curiously eyeing eye creams for when I can slather something on my dry under eye, especially something with spf if I can't find sunscreen that I can put near eyes. On the bright side a full face with lipstick and no eye makeup is certainly a Look and I'm kind of digging it.
Wearing: