Search.php still loads in the background before the Jetpack Search modal loads

### Impacted plugin

Jetpack, Search

### What

When using a Productivity App - search.php loads in the background before the Jetpack Search modal loads

### How

Sometimes when you have JP Search enabled on the site, the search.php file loads in the background while you wait for the JP Search modal to pop up. This is especially evident when using a Productivity App (like Alfred).

As this page isn't always styled, it would be great if this didn't happen. It would be better to have a blank page, than an un-styled one.

Here's a screen recording showing (what I think is) search.php before the Jetpack Search Modal displays.

https://github.com/Automattic/jetpack/assets/426388/6b44aa34-e9e7-4e5c-8b0d-a79d9f503280

Is it possible to change this, so if JP Search is enabled, the Search.php file doesn't ever load?

Start by reproducing the behavior with Jetpack Search enabled, using a productivity app such as Alfred, and observe the search.php request before the Jetpack Search modal appears. Trace the Search loading path around search.php and the modal; done means search.php no longer loads in the background and the modal opens without displaying an unstyled page.

Written by the indexing model from the issue text.