I have been unable to download free games on the App store without it requesting billing verification

If any of the six options listed below is true, selecting "None" as a payment method won't appear, and you cannot remove an expired or lost card. In such cases, providing billing information is mandatory.

The "None" option will not appear...

1. If you have an unpaid balance or payment due
2. If you have a subscription that automatically renews
3. If you're using the iTunes Store or App Store for the first time
4. If you are not in the country or region you’re creating an Apple ID for
5. If you changed your country or region
6. If you're a part of Family Sharing

Based on your country of residence and country you have selected under your Apple ID --> Payment methods that you can use with your Apple ID - Apple ...

Have you a Debit/Credit card issued to you that is valid internationally and from the same Country / Region as your Apple ID?

Know it here --> If your payment method is declined in the App Store or iTunes ...

Check the following

1. The payment method that you are adding is of the same country as that of App Store Country on your iPhone
2. You have a valid payment plan debit/credit card from an authorized bank from the same country
3. Payment methods that you can use with your Apple ID - Apple Support
4. Learn more about family purchases and payments.
5. Please check if your Debit/Credit card is valid internationally