why will my phone not let me delete my payment method

Because you may need to have a payment method on file, even if the app you want to download is free ->

Check which payment method you have on file 

You can also use your Apple ID balance as payment method: Add money to your Apple ID balance

Add money to your Apple ID balance – Apple Support (UK)

See also -> Payment methods you can use with your Apple ID

Payment methods you can use with your Apple ID – Apple Support (UK)

Hey there,

See ☞ Change, add, or remove Apple ID payment methods - Apple Support

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See ☞ If you can't remove a payment method from your Apple ID - Apple Support