Hi,
I am a beginner trying to learn Shopify development (coming from more WordPress/WooCommerce/PHP) and was practicing this very thing a few weeks back in Dawn Theme on my development store. I think your input fields need the form=“product-form-{{ section.id }}” attribute in them. I believe when going to checkout, this is how the data from your custom fields gets to the checkout page. Again, I could be wrong but it might be worth a try. I’m a newb with Shopify too. The section.id portion of the code might be specific to just Dawn theme too so I am not sure. Hope this gets you going in the right direction at least. Good luck!