Stop doing useless prettier-related work when doing a project search.

Before, project search might cause

<img width="1728" alt="not_pretty"
src="https://github.com/user-attachments/assets/5f8b935f-962d-488e-984f-50dfbaee97ba"
/>

but now we debounce the prettier-related task first, and actually set
the "installed" state for the default prettier, when there's no install
needed.

Release Notes:

- N/A