Have you gone back into your settings and checked it out? It disappears after I click it but reappears after going out of the settings and back in.
Yes, I have experienced the same issue. Clicked the Make Default at Apple button (after which the button is no longer visible in the Wallet app), received an automated response from Apple with the subject line Your Apple ID information has been updated, but if I go back into the Wallet app, the Make Default at Apple button has returned. You can even click it again and get the same automated response from Apple. So I would assume this is just a temporary bug.
FWIW, I was finally able to apply for the Apple Card just by going into the Wallet app and hitting the big Plus (+) button at the top right, where I was finally able to select the Apple Card, which I had done repeatedly starting yesterday. My advice is just to be patient, because it's not going to work until Apple finally enables your unique Apple ID in whatever random order they're using.
Also, that was at 3:43 PM Eastern Time (I'm on Long Island). Around 5:32 PM Eastern Time, I finally got the email invite from Apple to apply for the Apple Card