My wife's Android is newer than mine. I still have the old version of the app on my Android, but hers got updated the way described above. There is "Start in App" (to finish in a physical store) and "Pay online" option. I tried the latter and logged into her MoneyGram account (I had set up that account for her a while ago). Then I found "American Express - next day", and entered the amount, but for some reason it's impossible to enter the biller's account number. I also tried adding a VGC as a way of payment but it didn't go through.
Please let me know if anyone managed to pay a CC using that new interface.

Note also that this new "solution" implies that even if it's possible to pay a bill online it can only be via MoneyGram, not CheckFreePay.