I'll try to answer in 2 ways, since you gave your own "Scientific Method in a nutshell" definition:
1 - As you put it, the fact that causation is central to the objective of SM ends up putting Science and Philosophy nearer than what seems to be your vision given the title question, since "causation" and "why" are connected ideas.
Seeing things this way, I'd say that yes: a philosophical explanation about something testable will get the benefits of SM, although maybe it's trickier to rule out concurrent philosophical theories by experimentation.
About the why and how thing, I don't think it's a good way to understand the difference between Philosophy and Science, but assuming it is, I suppose that "how" is more tightly linked with the data. "Why", in its turn, admits different ways of explanation. This would, in fact, mean that the trustworthiness of SM would be weaker when applied to Philosophy.
2 - The wikipedia¹ takes the Oxford English Dictionary definition of SM to be:
"a method or procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses."
If we give up the How × Why distinction, I'd say again that yes: we can apply SM to Philosophy where all (or at least most) of necessary parts are present. If philosophy rarely applies full SM, it's because the studied subjects are more speculative and difficult to "square" in the SM framework.
About the trustworthiness, I don't see any problem "by definition", although if we don't have all the component parts of the method, we'll not get it to exert its full power.
One should have in mind that this kind of question is always puzzled with the problems of defining what is philosophy. Maybe some of the appealing character of this question comes from the fact that some people who try to define philosophy take as one of the basic characteristics the non-experimental factor, so this opposition (with which I don't agree) emerges.