What all of these use-cases have in common is the fact that the state is private

Suppose that all class and struct members which were declared private, were always mutable, even if the class instance is const.

That would mean you could do this:

class RandomNumberGenerator;
{
     private long seed;

     const int rand() /* guarantees not to modify any non-private member variables */
     {
         seed = f(seed);
         return (seed >> 32);
     }
}

const RandomNumberGenerator rng;
writefln(rng.rand());

In C++, we would have declared seed "private mutable" to achieve the same thing, but in D we tend to assume that everything in the same module is "friendly", so why not just let "private" mean "private mutable"?