I mentioned the delay, simply because this has happened a few times when I had a different card selected. Usually, when the process works correctly (select the card, hold it over the spot at checkout, double-click), there's a very brief delay and you get a message on the checkout terminal that might say something like "Waiting for Cashier" or something. When the AMEX card was used by Apple Wallet (without my selection) there has been a bit longer of a delay.
The AMEX card was never my default card. A bank Visa card has always been my default card. I deleted my AMEX card by going into Wallet & Apple Pay under settings on my phone and clicking "Remove Card."
To reiterate, I initiate the transaction in the grocery store by tapping on Apple Wallet, selecting the card I want to use, holding it over the designated area on the store checkout terminal, and double-clicking the side button when prompted.