Widgets REST API: Deleted widgets re-appear after accessing old widgets screen

[noisysocks]

From ​https://github.com/WordPress/gutenberg/issues/31575.

## Description

The REST API endpoint for deleting widgets does not seem to properly delete widgets.

## Step-by-step reproduction instructions

For making REST API requests, I use ​https://httpie.io and have the Basic Auth WordPress plugin installed, but use whatever suits you.

1. Install the Classic Widgets plugin.
2. Navigate to wp-admin/widgets.php.
3. Add a widget.
4. Find the widget's ID by hitting GET /wp-json/wp/v2/widgets.
5. Delete the widget by hitting DELETE /wp-json/wp/v2/widgets/{id}?force=1.
6. Hit GET /wp-json/wp/v2/widgets. Notice the widget is gone.
7. Navigate again to wp-admin/widgets.php. Notice the widget appears in Inactive widgets.
8. Hit GET /wp-json/wp/v2/widgets. Notice the widget appears again.

## Expected behaviour

The widget should not appear in steps 6 and 7.

## Actual behaviour

The widget re-appears.

The DELETE /wp/v2/widgets/:id?force=1 endpoint was only removing widgets from the 'sidebars_widgets' option and not actually deleting the widget's options from the "widget-$id_base" option. This PR implements widget deletion.

https://core.trac.wordpress.org/ticket/53313

[noisysocks]

In 51059:

REST API: Fix delete widget endpoint

Makes the DELETE /wp/v2/widgets/:id?force=1 endpoint actually delete the
widget from the "widget-$id_base" option and not just remove it from
'sidebars_widgets'.

Fixes #53313.
Props TimothyBlynJacobs.