I like the way this is handled in Go with the defer keyword: https://blog.golang.org/defer-panic-and-recover
This construct gives you most of the power of C++ RAII without the overhead. Except that you can't use it to cleanup resources after exiting an anonymous block—it strictly defers to function exit.
But it leaves out the most useful part: in C++ releasing the resources is completely implicit:
{
std::ifstream in(fn);
// Do something with 'in'
} // Released
Sure, it's an improvement over languages where you have to cover every possible scope exit, but it definitely doesn't give you 'most of the power' of C++ RAII.
In all other cases (which is almost all usages), it is semantically equivalent to RAII, so the language doesn't force any runtime overhead. The only difference is that the compiler is less mature than an average C++ compiler, but this is an implementation problem, not a design problem.
RAII has no overhead because it is a pattern designed within the context of a zero-overhead language. defer allows you to implement a superset of cases that RAII handles, including those with runtime overhead.
I hadn't considered the type of overhead you're talking about, which is admittedly more important for the kind of software that tends to get written in C.