Why can't the payment method be removed and added even after canceling the subscription?

The way to do it is to add the new payment method first, then you can remove the other payment method.

If you are looking to remove all payment methods, then along with cancelling the subscription, you will need to contact Apple directly to remove the subscription immediately, since they are still active until the end date of the subscription. Cancelling the subscription just prevents it from being renewed, but it remains active and to remove all cards there cannot be any active subscriptions.

Contact Apple for support and service - Apple Support