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Imagine an omnipresent and homogeneous field, let's say an infinite field of corn. By looking at the corn, you can see how fast you're going, without that observation requiring the field to have a center.

This is a great analogy, finally “clicked” for me after trying to understand this concept for a while. Thank you.

They are coming at you from every direction at once.

What you’re asking is analogous to asking where the center of the Earth is on a map of the surface, and arguing it must exist because of how things on the horizon change when you move.

The reference velocity is the one such that you see an equal amount of cosmic background radiation in every direction. If you're moving relative to it, you'll see extra radiation from in front of you (much like how you see extra cars moving in the opposite direction from you on a highway).