Isn’t rest_do_request() filtered by pre_get_posts() filter?

Removing the conditionals for the set() calls will solve the problem. I suspect that the REST request is populating those parameters, and as such the set() calls do not occur.

Untested:

add_action( 'pre_get_posts', 'test_pre_get_post_filter' );
function test_pre_get_post_filter( $query ) {
    if ( is_admin() ) {
        return;
    }

    $query->set( 'post_status', array( 'publish' ) );
    $query->set( 'posts_per_page', 7 );
    $query->set( 'orderby', 'title' );
    $query->set( 'order', 'asc' );
}