Search: new filters added via the block widget editor are not shown in search modal

### Impacted plugin

Jetpack, Search

### Steps to Reproduce

1. Create a new site and install the Search plugin.
2. Activate a Search plan.
3. Head to `/wp-admin/widgets.php` and add a new filter to the Jetpack Search sidebar.
4. Visit the front end search modal at `?s=` and check the displayed filters

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

If I configure filters in the widget editor, I expect them to appear on my site's front end.

### What actually happened

The configured filters do not appear.

### Other information

@kangzj has deduced: "for a site that never used a theme with sidebars, now legacy widgets are stored as block widgets"

### Operating System

_No response_

### OS Version

_No response_

### Browser

Firefox

### Browser Version(s)

_No response_

Reproduce the issue by configuring a Jetpack Search filter at /wp-admin/widgets.php, then open the front-end search modal at ?s=. Trace how the configured filters move from the block widget editor to the modal; done means the added filters are displayed on the front end.

Written by the indexing model from the issue text.