The main issue in your code is that $product_cat_id is not defined. Also, it is better to display the button after the existing link surrounding the thumbnail and the title.
Try the following instead:
add_action( 'woocommerce_after_subcategory_title', 'view_category_button', 20 );
function view_category_button( $category ) {
printf('%s', esc_url( get_term_link( $category, 'product_cat' ) ), esc_html__( 'View more', 'woocommerce' ) );
}
Code goes in functions.php file of your child theme (or in a plugin). Tested and works.