What do you do when your normal programming language stops being your normal programming language (you add a language, switch languages), but you want to preserve details of the configuration? Translate it all into your new language(s)? How do you keep these things synced up? You extract the details into a configuration language/serialization format and then deserialize in every language you work with.
Embedding your configuration in your application language only works well as long as you have one application language. (Or perhaps you use Lua or TCL which are both almost trivially embedded into every other language.)
You can compile your general purpose language DSL to json and then I think this isn’t too much of a problem.
A bigger issue imo is packaging. General purpose languages aren’t typed well-designed for producing single, understandable standalone files like config languages are. Like I think a simple config DSL in TypeScript could potentially be a perfect way to solve this problem, except that no one wants to lug around a package.json, package-lock.json, and node_modules directory just to write a bit of config. Yet the ability to bring in npm modules—especially those containing relevant types—is where a lot of the attraction of using TypeScript comes from.
So turn the general purpose language into a non-general purpose language by creating a compiler for a subset of it. That might be what we'd call a configuration language then since it's no longer actually the original language (in full). Like JSON and others.
Is that a real problem to be solved? Do you switch languages in an existing project that often? When you do, do you just copy/paste your config files or do we have to rebuild them in whatever config language the new language uses?
Embedding your configuration in your application language only works well as long as you have one application language. (Or perhaps you use Lua or TCL which are both almost trivially embedded into every other language.)