DaveTheCoder Thanks for all your help so far.
I first downloaded your project and APK, and tried installing the APK on my devices. Strangely, it failed to even so much as install on either of them.
The project loaded fine in my Godot editor (those warnings seem to happen a lot when importing projects - I've been getting them the whole time I've been using the Godot 4 beta) and I was able to test it on my PC. Exporting an APK and installing it onto my Android devices worked, but the same as with my own projects, running it didn't - I had the same black screen on both. As before, on the Retroid, the Godot splash was shown, then a moment of black screen, and then there was a crash.
This is still helpful, though, as it's helped to narrow down the issue. As far as I can see, either Godot 4 is currently exporting files that my particular Android devices - or perhaps versions of the Android software below Android 10 - don't like, or there is something horribly wrong with my Godot settings or installation. But, if the latter is the case, I find it strange that your APK refuses to install on my devices, and that my APK installs and runs just fine on yours, so I'm inclined to believe the exports are being picky about which devices or Android versions they want to play ball with.
In short, I'm pretty sure at this point that it's just a compatibility issue.
I'm not sure we're going to be able to get any further unless somebody who knows a lot more about Godot 4's Android compatibility finds the thread and chips in. For now, I'm just going to soldier on with my project, accept that I won't be testing it on my Retroid any time soon, and hope future versions of Godot 4 come with better Android compatibility (or someone is able to work out what's wrong here, if it's not that).
Again, thanks, Dave, for all of your help.