Why does Apple not allow AppleCare to be bought online using Apple Pay or Apple Balance?

Hello.

I am trying to do a simple transaction: buy AppleCare+ for my iPhone 13 Pro Max.

I have tried doing it through

(1) my phone -> Settings -> General -> About -> AppleCare coverage available -> 24 months coverage

(2) my Mac -> Safari -> Apple.com Support by checking the phone's eligibility, running diagnostics, and going to 24 months coverage.

I have more than enough Apple Balance (redeemed gift cards) and a couple of cards that I got recently.

I have Apple Pay activated (have been using it almost since it was available) and a Visa card is tied to it. It works wherever normally Apple Pay would work. Even the Apple Balance is a payment method listed on there to be used as Apple Pay.

I have PayPal as a third payment method.

All these three payment methods are linked to my iCloud settings.

When I try to buy AppleCare through the two methods listed above, none of these three payment methods show up. I am continuously asked to link a new credit card. I put up the gift card in front of the camera and nothing.

I have used Apple Balance and Apple Pay to buy stuff from Apple before, including AppleCare. I bought an iPhone SE before and bought the AC with it - and paid for everything with Apple Balance. I pay for the post-24 months AC on my iPad through Apple Balance. I have bought stuff from the Apple Store using Apple Pay. What's the holdup now?

It seems like the issue was there in 2020 as well: Buy AppleCare using apple gift card or ap… - Apple Community. Clearly, Apple hasn't fixed its coding yet on this. Maybe either too few people buy AC after buying the device instead of buying it right away, or they buy the monthly plan.

I have been on the phone with Apple Support for 30 minutes and I was transferred to three different agents. Nobody could figure out the problem with Apple Balance or Apple Pay and just said, "you should add a payment method." I mean, I could have done that, but what's the point of having a digital wallet if it doesn't work at Apple's own store?

What am I missing? Is there a secret code that I should know and I don't?