Thanks for mentioning this trick with figuring out CODE_SIGN_IDENTITY, @m-miscampbell! Unfortunately, I could not get it working because it looks like it conflicts with automatic code-signing.

Same story - I do see the button for one app and I don't see it for another app. Both apps are built with Xcode 13, as far as I know. Wondering if it might be related to difference in the build settings or to the fact that the app that has no download button employs Fastlane for upload to TestFlight.