Search: Jetpack Search widget preview is incorrect in the block-based widget editor

### Impacted plugin

Jetpack, Search

### Steps to Reproduce

1. Purchase a Jetpack Search subscription with either the Jetpack or Search plugin on a site with WordPress 5.9 or higher.
2. Navigate to the widget editor `/wp-admin/widgets.php`.
3. If one does not already exist, add a Jetpack Search widget to the "Jetpack Search Sidebar".

### A clear and concise description of what you expected to happen.

For the Jetpack Search widget preview to be similar to what's actually shown on the site.

### What actually happened

The widget preview looks like the following:

While this is what's actually shown on the site:

### Other information

_No response_

### Operating System

_No response_

### OS Version

_No response_

### Browser

Chrome/Chromium

### Browser Version(s)

_No response_

Reproduce the issue on WordPress 5.9 or higher with a Jetpack Search subscription, then open /wp-admin/widgets.php and add the Jetpack Search widget to the Jetpack Search Sidebar. Compare the block-based editor preview with the rendered widget on the site; done means the preview is similar to the live appearance.

Written by the indexing model from the issue text.