It should work, given what you tell us. There may be something you haven't told us which would explain why it isn't. You are seeing no error messages? Often the case is similar to this:
If you can't remove your last payment method or use no payment method with your Apple ID - click on this link to "Learn why you can't use your Apple ID without a payment method in the App Store, iTunes Store, and more." --> https://support.apple.com/HT203905
#1 in the last link is a frequent cause, where somebody has a subscription which means you cannot change the payment method. You must wait for the subscriptions to expire, or contact Apple and see if they can resolve the issue.
This is the most direct method to contact Apple for media/app/iCloud store support:
iTunes [Store] Support - Use this form for assistance with account and billing, code redemption, or purchase questions -->
https://www.apple.com/emea/support/itunes/contact.html
Use this method to get telephone support:
Contact Apple for support and service - "See a list of Apple phone numbers around the world."
https://support.apple.com/HT201232