Instant Search: Private simple site search fails if mapped to a domain

This issue will only occur if you disable third-party cookies.

#### Steps to reproduce the issue

1. Create a private WordPress.com site.
2. Map it to a non-WordPress.com domain name (e.g. example.com).
3. Purchase and enable Instant Search.
4. Do a site search.

#### What I expected
Search overlay to appear and be populated with results.

#### What happened instead
Search overlay appears and remains stuck in its loading state.

#### Context
This is likely due to the browser counting `public-api.wordpress.com` as cross-origin/third-party when on a mapped domain.

Reproduce the Instant Search failure on a private site mapped to a non-WordPress.com domain with third-party cookies disabled. Start at the search overlay loading flow and its request to public-api.wordpress.com. Done means the overlay finishes loading and displays search results in this configuration.

Written by the indexing model from the issue text.