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I agree about the old answer, but it didn't occur to me at first and I guess it's too late to change it now. And yes, I believe that you are right about the action.
Take a look at Proposition 2.6.11 in Engelking's General Topology. I'm not sure if this answers your question since he uses different nomenclature and I have no time to sort it right now, but it is certainly related.
Take a look at Sections 4 and 5 of Jardine's Cubical Homotopy Theory: A Beginning. I doesn't quite give the definition that you want, but maybe it comes close.
If I'm reading the notation of the question correctly, we are given morphisms $X \to Y$ and $X' \to Y'$, not the other way round. I don't think that this is necessary. For example, you could have a morphism that is a composite of two: one satisfying this condition and the other satisfying the symmetric condition. Such composite wouldn't necessarily satisfy either.
The answer below already explains that this doesn't hold. However, there is an easy sufficient condition. If in addition to $f$, $g$ and $h$ having the LLP with respect to $S$, we assume that the induced morphism $Y \sqcup_X X' \to Y'$ has the LLP with respect to $S$, then the answer is positive.
I don't know if this helps with your $TC$ question, but this paper arxiv.org/abs/math/0601221 sets up some foundations of Goodwillie calculus for non-finitary functors.
Arguments of May and Joyal are combinatorial and very explicit while the argument of Gabriel and Zisman is categorical but still quite explicit. In the grand scheme of things they amount to the same. Of course, for some specialized purposes the combinatorial argument will be more appropriate.
To be fair, I don't think it is accurate to say that this approach avoids "the theory of anodyne extensions". In both cases, we prove that the pushout product of $\partial \Delta[m] \to \Delta[m]$ and $\Lambda^{k}[n] \to \Delta[n]$ is anodyne which I is pretty much "the fundamental lemma of the theory of anodyne extensions".
