It might be that your profile doesn't have access to the Xcode Cloud development certificate under 'Users and Access' in App Store Connect.

I figured out the issue. The build number is incorrect, because Xcode Cloud sets it itself, even if you define it yourself. According to this document, it always starts from (1) and increments. So, if you have an existing build in TestFlight (say version 1.0.0) and have previously submitted builds and manually incremented the build number, it might be at, say build (10). Then when you start your Xcode Cloud builds it just defaults to build (1) and increments from there, failing when it gets to the 'submit to App Store Connect' stage because the build number it's decided on (1) – regardless of what you've set in the project – is lower than the build for the existing version (10). You can either keep running builds until you get to the build number already in TestFlight 🤨 or bump your version (e.g. to 1.0.1) and App Store Connect resets its expectations for build numbers.

So looks like there's an internal error, according to the more verbose logs. However, weirdly I managed to build and upload it manually with the build number of 19, and this is now finally showing up.

Apologies, I meant 'Xcode Cloud' in the Archiving stage: The Identity screen is under the 'General' tab in the Target. The current release version in ASC is 3.0.1. As you can see from the Fastlane screenshot above, the current TestFlight version is 4.0.0. It should allow me to upload a build with number 20.