Publicize: move publicize data to the store

Currently, when a publicize request is performed by the client, the relevant data used to track the request/response flow is stored in a hook state.
Let's move this data into the `jetpack/publicize` store.

Start by tracing the client-side publicize request and the `jetpack/publicize` store. Identify the hook state that tracks the request/response flow, then verify that the store owns this data and the publicize flow still works without that hook state.

Written by the indexing model from the issue text.