Does "Thomason-contractible" mean "with a contractible nerve"? If so this is not sufficient and my previous comment explains why. There are contractible direct categories with no initial object, for example the direct part of $\Delta$ (i.e. the category of finite nonempty totally ordered sets and injective order-preserving maps). There are also examples among posets.

If $P$ is a direct category, then projectively cofibrant diagrams coincide with Reedy cofibrant diagrams. It is easily seen that only very uncomplicated direct categories will satisfy your property. For example a constant diagram is always a "diagram of cofibrations", but constant diagrams are Reedy cofibrant only when $P$ is a coproduct of categories with initial objects. The situation is probably only worse for non-direct categories.

...This is exactly the meaning of the word "natural" in the categorical context: transformation between functors defined on some category $\mathcal{C}$ is natural when its definition depends only on properties shared by all objects of $\mathcal{C}$. This is of course subjective, but it corresponds pretty well to my own informal understanding of the word "natural".

I disagree that this last transformation is not "natural" in informal sense. There is one such transformation for every polynomial with integer coefficients and the only "choice" we make is which polynomial to use. In other words we choose which transformation we define, but there is no "choice" involved in the definition of the transformation itself. The point is that this definition is uniform in the sense that one formula works equally well for an arbitrary ring $R$ and does not depend on any of its intrinsic properties...

The mentioned proof in "Simplicial objects in algebraic topology" does use minimal complexes. Is there also a proof of this fact that avoids them?