why is std::basic_ostream::operator<< not const-qualified?

First, an example to illustrate the morals behind my question: the below code will not compile, because std::basic_ostream::operator<< is not const-qualified. (https://gcc.gnu.org/onlinedocs/libstdc++/libstdc++-html-USERS-3.4/ostream-source.html shows that the operator is not const-qualified.)

I compiled with GNU g++ 6.4.0 compiler, with --std=c++11 flag on.

#ifndef TEST_H
#define TEST_H
#include<string>
#include<iostream>
using namespace std;
class ChessPiece{
    const string name;
    const int side;
public:
    ChessPiece(const string&,const int);
    void printPiece(const ostream&) const;
};
#endif // TEST_H

...and test.cpp.

#include"test.h"
ChessPiece::ChessPiece(const string& s,const int bw): name{s}, side{bw} {}
void ChessPiece::printPiece(const ostream& S=cout) const{
    S << "a " << (side==1?"white ":"black ") << name << endl;
}
int main(){
    ChessPiece p{string("pawn"),-1}; // a black pawn
    p.printPiece();
}

However, I am not sure as to why these kind of errors should occur in the first place, even though operator<< is, as in the above code, logically const. Yes, the obvious answer would be that "operator<< changes the inner state of std::ostream in some way".

However, I know that by making the members mutable we can change the contents of the class, as long as the const-qualified function is logically const. I also know that an instance of std::ostream will not logically differ in any way before and after calling its operator<<. (If anything I wrote is wrong, please point it out. Thank you)

Rephrased,

Why is the logically const std::basic_ostream::operator<< not const-qualified, instead of having some of its members mutable?

Thank you in advance.