No, reflection doesn't break the idea of private methods. At least not per se. There is nothing that says that reflection can't obey access restrictions.
Badly designed reflection breaks the idea of private methods, but that doesn't have anything to do with reflection per se: anything which is badly designed can break the idea of private methods. In particular, a bad design of private methods can also obviously break the idea of private methods.
What do I mean by badly designed? Well, as I said above, there is nothing stopping you from having a language in which reflection obeys access restrictions. The problem with this is that e.g. debuggers, profilers, coverage tools, IntelliSense, IDEs, tools in general need to be able to violate access restrictions. Since there is no way to present different different versions of reflection to different clients, most languages opt for tools over safety. (E is the counterexample, which has absolutely no reflective capabilities whatsoever, as a conscious design choice.)
But, who says that you cannot present different versions of reflection to different clients? Well, the problem is simply that in the classical implementation of reflection, all objects are reponsible for reflecting on themselves, and since there is only one of every object, there can be only version of reflection.
So, where does the idea of bad design come in? Well, note the word "responsible" in the above paragraph. Every object is responsible for reflecting on itself. Also, every object is responsible for whatever it is that it was written for in the first place. In other words: every object has at least two responsibilities. This violates one of the basic principles of object-oriented design: the Single Responsibility Principle.
The solution is rather simple: break up the object. The original object is simply responsible for whatever it was originally written for. And there is another object (called a Mirror because it is an object that reflects other objects) which is responsible for reflection. And now that the responsibility for reflection is broken out into a separate object, what's stopping us from having not one, but two, three, many Mirror Objects? One that respects access restrictions, one that only allows an object to reflect upon itself but not any other objects, one that only allows introspection (i.e. is read-only), one that only allows to reflect on read-only callsite information (i.e. for a profiler), one that gives full access to the entire system including violating access restrictions (for a debugger), one that only gives read-only access to the method names and signatures and respects access restrictions (for IntelliSense) and so on …
As a nice bonus, this means that Mirrors are essentially Capabilities (in the capability-security sense of the word) for reflection. IOW: Mirrors are the Holy Grail on the decade-long quest to reconcile security and runtime dynamic metaprogramming.
The concept of Mirrors was originally invented in Self from where it carried over into Animorphic Smalltalk/Strongtalk and then Newspeak. Interestingly, the Java Debugging Interface is based on Mirrors, so the designers of Java (or rather the JVM) clearly knew about them, but Java's reflection is broken.