It seems that it is more sensible to say that entanglement is locality. The notion of two points being close to each other appears to be a course-grained approximation of high amounts of entanglement in the vacuum. If you divide the space along a boundary and break the entanglement between regions on both sides, the effect is to increase the distance between them. So perhaps it is better to think of gravity, locality and dynamical spacetime, as the "hydrodynamics of entanglement"

[0] and [1] are a public lecture, [2] is colloquium

[0]: https://www.youtube.com/watch?v=OBPpRqxY8Uw

[1]: https://www.youtube.com/watch?v=uiG_EtVQu5o

[2]: https://www.youtube.com/watch?v=PwAKr-h6kAI