> You can still break up your algorithm into a class with multiple template methodsTemplate methods are an abomination. They are only widely used because many OO languages don't have first-class functions. A template method means that the method is being parameterized by one or more functions. Parameters should be expressed as parameters! You wouldn't design a Circle class so that you need to subclass it to specify the radius of the circle. The radius is a parameter. Likewise for the callback functions used by a template method. Template methods are intrinsically difficult to understand and spaghetti-like, unless well-documented, because it is not immediately clear what the parameters to it are. And when overriding a callback method, it's not immediately clear what template method the overridden method is paramaterizing. Or that it is even parameterizing anything. (Caveat: Sometimes template methods are warranted, but they are greatly overused.)
> Classes have another advantage: virtual method calls.
You don't need virtual method calls in order to parameterize functionality. See previous paragraph. Furthermore, it is not safe to override a method without knowing if it has been designed for overriding, and what the class expects of the overriding method. See Effective Java, "Item 17: Design and document for inheritance or else prohibit it". Also see "Item 16: Favor composition over inheritance", and a similar discussion in the Gang of Four book.
> If one function in a module calls into another function in the module I can only do two things: a) copy/paste the library and change one of the functions or b) a monkeypatch which modifies a shared resource someone else might want to use with the old semantics.
If a method is not designed to be overridden, you shouldn't override it anyway. (See previous paragraph.) When using a functional approach (in a language with first-class functions), you can usually design for flexibility just as easily or more easily than when using an inheritance-based OO approach for flexibility.