Reason for that is https://github.com/piwik/piwik/blob/3.x-dev/plugins/WebsiteMeasurable/MeasurableSettings.php#L257-L259

Seems somehow to be on purpose, that the value isn't saved.
@tsteur any reason for that? Or can it simply be removed?

It is never saved because there is a condition defined which is supposed to determine whether it should be activated or not. I presume this was kind of 2.X behaviour. Possibly it should kind of "clear" the query parameter value when unsetting it but don't think this is possible just yet in JS directly. Therefore we would need to unset in MeasurableSettings.php a query parameter value when "default search" is activated.
This might be possible like

$field->transform = function ($setting, $settings) {
if ($setting->getValue()) { 
    $settings->getSearchQuery...()->setValue('');
}
                return null;// never actually save a value for this
            };

I haven't tested it as I am on Piwik 2 currently but something like this could work