On a nonconst object, why won't C++ call the const version of a method with public-const and private-nonconst overloads?

The object is not const, so the non-const overload is a better match. Overload resolution happens before access checking. This ensures that overload resolution is not inadvertently changed by changing the access of a member function.

From the standard:

13.3.3 If a best viable function exists and is unique, overload resolution succeeds and produces it as the result. Otherwise overload resolution fails and the invocation is ill-formed. When overload resolution succeeds, and the best viable function is not accessible (Clause 11) in the context in which it is used, the program is ill-formed.