I just want to note that rarely will just pointing to a tech note resolve the issue. I'd predict that usually if somebody posts here, they've already looked at the existing documentation. I know that's been the case for me. It's quite discouraging to hit some case that you can't tell why it's not working as documented, then have nowhere to turn for more specific help.

When I add this: <link rel="apple-touch-icon" sizes="180x180" href="/apple-touch-icon.png"> I see that the icon does appear on the "Info" sheet. I never looked there because I was expecting it on the main sheet. Was that an incorrect understanding? Is there no way to get it to appear on the main sheet, without having to click the circle-i button?

You can see above, though, that I did do this. In the two sizes specified in the linked document. Are you saying that instead I need to specify a 48x48 one?

What else can be checked? Again, this has nothing to do with our website, as the official Apple Pay demo site doesn't show the button either.

I don't think any of that addresses this case, where even the demo site isn't showing the button, but the cards are clearly in the device's wallet.

Gmail account worked fine for me as well. I'm surprised there's apparently no way to use an iCloud Apple ID? It's catch-22 that the Sandbox Apple ID has to not exist yet, but you can't create an iCloud email until the Apple ID already exists.