I don't have enough intuition to answer the OP's question, but I can answer this one. I'll answer from a different perspective.
First, consider an explosion. At any point in time after the explosion, the shrapnel is flying outwards from the center of the explosion. If you pick a piece of shrapnel and follow it, you'll see that pieces next to it are not really moving away from it very fast, whereas pieces on the other side of the center are moving away from it very quickly.
Now let's go back to space. If the universe was expanding outwards from a center, then you would be able to observe differences in the large scale motion of galaxy clusters depending on which direction you look. But when you look out, you find that everything is moving away from you in the same manner on large scales.
That seems to imply that your galaxy is the center of the universe. But every galaxy sees the same thing, and so every galaxy would conclude that they are the center of the universe. So there are 'centers' all over the place, which isn't really compatible with the idea of a center.