Jetpack Search: when using full-width overlay without overlay widgets, it's impossible to clear any pre-applied filters

#### Steps to reproduce the issue

1. Enable Jetpack Search, but without any widgets in the overlay.
2. Add a Jetpack Search filter widget to the main content of the page.
3. Apply a filter and enter a search query.
4. The overlay opens, but without any way to remove the applied filter.

#### What I expected

Some way to clear the filter - perhaps shown underneath the search box with a 'x' to remove each filter?

#### What happened instead

There is no way to clear filters inside the overlay.

Reproduce the issue with Jetpack Search enabled, no overlay widgets, and a filter widget in the page content. Start by tracing how the overlay search box receives the applied filter, then provide a visible way to remove it; done means every pre-applied filter can be cleared from the overlay.

Written by the indexing model from the issue text.