Restrict Search Query To After Specific Date

Note this problematic line:

$query->set( array( 'date_query' => array(

it should be

$query->set( 'date_query', array(

So try instead this form:

$query->set( 'date_query', 
    [
        [
            'after'     => 'January 1st, 2013',
            'inclusive' => true,
        ]
    ] 
);