An important point not mentioned by the article is that of "co-recursion" with inheritance (of implementation). That is: an instance of a subclass calls a method defined on a parent class, which in turn may call a method that's been overridden by the subclass (or even another sub-subclass in the hierarchy) and that one in turn may call another parent method, and so on. It can easily become a pinball of calls around t…
With interface-inheritance, each method is providing two interfaces with one single possible usage pattern: to be called by client code, but implemented by a subclass.
With implementation-inheritance, suddenly, you have any of the following possibilities for how a given method is meant to be used:
(a) called by client code, implemented by subclass (as with interface-inheritance) (b) called by client code, implemented by superclass (e.g.: template method) (c) called by subclass, implemented by superclass (e.g.: utility methods) (d) called by superclass, implemented by subclass (e.g.: template's helper methods)
And these cases inevitably bleed into each other. For example, default methods mix (a) and (b), and mixins frequently combine (c) and (b).
Because of the added complexity, you have to carefully design the relationship between the superclass, the subclass, and the client code, making sure to correctly identify which methods should have what visibility (if your language even allows for that level of granularity!). You must carefully document which methods are intended for overriding and which are intended for use by whom.
But the code structure itself in no way documents that complexity. (If we want to talk SOLID, it flies in the face of the Interface Segregation Principle). All these relationships get implicitly crammed into one class that might be better expressed explicitly. Split out the subclassing interface from the superclass and inject it so it can be delegated to -- that's basically what implementation-inheritance is syntactic sugar for anyway and now the complexity can be seen clearly laid out (and maybe mitigated with refactoring).
There is a trade-off in verbosity to be sure, especially at the call site where you might have to explicitly compose objects, but when considering the system complexity as a whole I think it's rarely worth it when composition and a tiny factory function provides the same external benefit without the headache.
These are powerful tools, if used with discipline. But especially in application code interfaces change often and are rarely well-documented. It seems inevitable that if the tool is made available, it will eventually be used to get around some design problem that would have required a more in-depth refactor otherwise -- a refactor more costly in the short-term but resulting in more maintainable code.