I read the checkout screen very carefully, and there was no information saying that I needed to fill in the card information for any other purpose than paying with it. Since I did not want to go through with a card payment, I saw no reason to fill in the card information there. However, your hint made me go to https://www.paypal.com/businesswallet/money and check my cards. They were expired, and after adding my new card information there, the checkout stopped at the balance option. Very, very weak information from PayPal, but many thanks to you @sharpiemarker that gave me the hint to find the problem!
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