I’m developing the same functionality right now and something like this happen to me (I was able to see the settings in the admin dashboard but not in the checkout page)…
What I did was check the option Enable debug mode in WooCommerce/Settings/Shipping/Shipping options and then I was able to see my custom shipping in the checkout page. After that I disable the option again and my shipping is still visible in the checkout page (I don’t know the reason but it works)
I was checking your code and notice that you’re missing an important action to add a custom shipping:
add_action( 'woocommerce_shipping_init', 'tutsplus_shipping_method' );
where the tutsplus_shipping_method is a function that defines the class that extends the WC_Shipping_Method.
The action woocommerce_shipping_init is the main action for WooCommerce Shippings which includes all the shipping classes before they get instantiated. By using that action we are sure that our shipping method will be included after WooCommerce has been initialized and in the right place for WooCommerce to use it.
I also notice that your filter add_filter('woocommerce_shipping_methods', array($this, 'register_shipping_method')); points to a function that doesn’t exist, maybe it’s a type error and you wanted to point to your register_shipping function.
I followed this tutorial and everything was great: https://code.tutsplus.com/tutorials/create-a-custom-shipping-method-for-woocommerce–cms-26098