Especially as the Adagio account page won't allow me to change the credit card that they have on file for me.
Which is why I haven't bought anything from Adagio for months
P.S. Shortly after writing this, I wondered to myself whether I had actually thought to notify Adagio support of this problem. So I contacted them via the website. They e-mailed me back within an hour to tell me that they'd deleted the credit card info on file and that I can now enter a new one. Fantastic service!
(Stilll, it might not hurt to offer PayPal as a supplementary option.)