I can't use my Apple Gift card as an Apple ID payment method

It is likely you need a valid payment method. This would be needed to verify your identity and also to validate that the payment method is in the same region as the device. Some subscriptions for services are only available in certain regions. With the valid payment method, the money will still be deducted from your account balance that you funded with the gift cards. The only time your payment method would be charged is if there was not enough money in your Account balance to cover the subscription.

I don't know any way to get around having a valid payment method to be able to download apps. With your Visa gift card used up, you would need to add another payment method and then you would be able to remove that card. I don't know why it would say you have active subscriptions when you have cancelled them, but would suggest you contact Apple directly.

Contact Apple for support and service - Apple Support

Note that simply cancelling a subscription doesn't mean it isn't active. It will stay active until it fully expires or you tell Apple to make it expire now.

I am surprised that you say you got a Vanilla card to work as a payment method. It is not a true credit or debit card and Apple won't accept other brand gift cards.