Apple won’t let me buy apple care with my gift card.

When redeeming a gift card, the amount will be credited to your Apple Account balance.

Looks like it is not possible to pay Apple Care with your Apple Account balance.

Apple Care is not listed here -> What you can buy with your Apple Gift Card or Apple ID balance

What you can buy with your Apple Gift Card or Apple Account balance – Apple Support (UK)

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Apple Gift Card and Apple Account balance can't be used for some purchases

• Some purchases might require you to have a credit card on file, even if you use your Apple Account balance to make the purchase.

for sure you can -> Call Apple Support at 1-800-APLCARE.

See also the following Apple articles:

Add a payment method to your Apple ID

Add a payment method to your Apple ID – Apple Support (UK)

Payment methods you can use with your Apple ID

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

fyi: Not all credit & debit cards seem to work with App Store / iTunes Store, try a different one, or preferably a credit card. If your card is not accepted as a paymenf method, you need to contact your bank.

If you still need help, get support from Apple directly -> Get Support or call Apple in your region see -> Contact Apple for support and service 

We are just other users here and do not work for Apple; we have no influence on their policies.You can contact them and ask, but it they do not allow payment with a gift card, it is their prerogative.