I see your point, but I 80% disagree with your conclusion.
When you test opaque objects, you're generally not trying to test individual state transitions, you're trying to test that the class's interface adheres to the external promises it makes.
That said, many OOP languages have solutions specifically for when you actually do need to test those internal state transitions. C# for example has the "internal" keyword and allows you to declare friend assemblies, so you mostly get your cake and eat it too, at the cost of not hiding the code from yourself as the module implementer.
I interpreted the top-level comment as a list of extra external promises you could add to your class's interface (incremental search, save/restore, etc). If those are easier to test with functions and a state parameter, it might be better to skip the class.
When you test opaque objects, you're generally not trying to test individual state transitions, you're trying to test that the class's interface adheres to the external promises it makes.
That said, many OOP languages have solutions specifically for when you actually do need to test those internal state transitions. C# for example has the "internal" keyword and allows you to declare friend assemblies, so you mostly get your cake and eat it too, at the cost of not hiding the code from yourself as the module implementer.