assistant_context_editor: Don't block `ContextStore` initialization on reloading contexts (#24052)

This PR changes the `ContextStore` constructor to not block on reloading
the contexts before we finish initializing it.

I noticed that the Assistant panel was taking a long time to show up in
the status bar, and upon further investigation uncovered that with a
large number of contexts (I have ~320) it takes a long time to load them
all.

Release Notes:

- N/A