There are different types of coroutines. The C++ type are sometimes called "stackless coroutines". With stackless coroutines you can't yield from a nested function call. Stackless coroutines are basically generators where you can pass arguments through resume, and async/await is effectively a form of stackless coroutines with yield/resume semantics that aren't fully generalized as coroutines, but oriented toward some bespoke notion of concurrency rather than as an abstract control flow operator.
"Stackful coroutines" allow yielding from any arbitrary point. They're basically fibers, except with the explicit control transfer and value passing yield and resume operators; there's no hidden or implicit control transfer like with green threads. Though, some people would argue allowing any function to yield without announcing this in their type signature is tantamount to hidden control transfer. Personally, I don't see how that's different than allowing any function to call other functions, or to loop, but in any event languages are free to layer on additional typing constraints--constraints that can be tailored to the desired typing semantics, rather than dictated by implementation details.
Stackless coroutines are typically implemented as a special kind of function whose state is allocated and instantiated by the caller. In contrast, stackful coroutines are typically implemented by reifying the stack, similar to threads. The "stack" may or not be the same as the system's ABI stack.
In stackful coroutines, unless there are additional typing constraints imposed by the language for hygiene reasons, any function can typically be called as a coroutine or use yield and resume. There's no need to compile functions into special alternative forms as call frame management works the same whether invoked from a coroutine context or not.