Andrej, thank you for your reading of May and Sigurdsson, which I wish we
had made more accessible.   I don't have time to check anything, but my 
recollection is that the verification of the observation is easy, whether 
or not it is formal, and works equally well in homological analogues.  I 
think we inserted it only for reassurance, since it is of course 
to be expected.  It is obviously true when spaces are restricted to be compactly 
generated since then there is no ``strong'' distinction. (MS could not just
restrict like that because of point-set problems with parametrized function spaces).

The paper of Schw\"anzl and Vogt noted in Karol's answer is the original source 
for the distinction between strong and ordinary Hurewicz cofibrations and fibrations 
and may well be helpful.   A crucial non-formal thing, noted just after the 
observation you refer to and emphasized more strongly in Schw\"anzl and 
Vogt is the need for a good cylinder object:  $\partial I \longrightarrow I$ has the LLP 
wrt $h$-acyclic $h$-fibrations.  (Of course, that fails for simplicial sets.)


In case you are thinking about model structures, I'll say a bit about that
(probably repeated from another comment or answer).  There is a mistake in the 
general proof of the factorization axioms in the paper of Cole that Karol refers 
to and in MS (4.4.2) and MP (``More Concise'').  A fix is given in: T. Barthel and E. Riehl. 
On the construction of functorial factorizations for model categories. Algebraic 
& Geometric Topology 13 (2013), 1089--1124.  (That is the paper referred to by Omar, 
but their fix does not seem directly relevant to the precise question you ask).   They and I 
are just finishing a related paper "Six model structures for DG-modules over DGAs'', 
in which their fix is again utilized and is described in full categorical generality. 
If anyone is interested, I will be happy to say why ``six'' and why that is also to be 
expected in topological contexts of interest (but I sometimes get scolded for digressing).