Search: toggling module on/off removed widget from sidebar

Steps:
- purchase a Search Plan (no other plan) and turn on search
- configure side bar and overlay widgets
- turn off search from the settings page
- see that the widgets have been removed from the front end of the site
- turn search back on
- see that the widgets are not back

Also weird:
- go to the wp-admin/widgets.php
- old widgets may reappear for some strange reason (in my case I got an old widget config that I used to have)

Reproduce the sequence from the Search settings page, including configuring sidebar and overlay widgets, then inspect wp-admin/widgets.php after toggling Search off and on. Confirm that existing widgets remain available on the site and that unexpected old widget configurations do not reappear in the admin.

Written by the indexing model from the issue text.