The problem is not that there are too many configuration languages. The problem is that none of them are any good. The way to solve that is to keep trying new ones until we find one that is good.
Go ahead and write your configuration as a fully Turing complete sub-application in whatever language you like. It will do everything you could possibly want, and in a few years it'll grow so complex and hairy that it will need its own fully Turing complete sub-application to configure it, lather, rinse, repeat. If you're really clever, every configuration layer will be written in a separate language with its own dependency tree, test framework, and toolchain.
Personally, I prefer not having to recompile just to change some variables and settings. I'm fine with INI or JSON (although I prefer Lua tables) when they're appropriate. The problem is not that configuration languages are a bad idea, the problem is interminable Turing creep and developers wanting every aspect of their applications to be as flexible and powerful as possible.
The premise of having a separate config language is fine - somewhere, somehow, inevitably, you're going to need a read-only data store for globals and references to system settings. You can hardcode all of those variables in your application or put them elsewhere.
I'm genuinely intrigued to see a non-trivial example of this, a configuration in use by an organisation with complex needs, describing its applications and its cloud infrastructure, while avoiding the pitfalls you describe that bedevil the use of Turing complete languages.