I have recently begun reading about model categories. In particular, I have been using Balchin's A Handbook of Model Categories as a reference, and the following quote has been quite perplexing.
A cofibrantly generated model category has a small set of cofibrations (resp., acyclic cofibrations) that generate all other cofibrations (resp., acyclic cofibrations). Many arguments are much easier to check under this assumption as conditions can be checked on the generating sets.
My question is as follows: what are some motivating examples of the aforementioned arguments that might be easier to check on generating sets? For instance, the classical model structure on simplicial sets can be realized as a cofibrantly generated model category, with generating cofibrations being the boundary inclusions and generating acyclic cofibrations being the horn inclusions. Are there arguments that are easier to check on these generating sets than it might otherwise be without invoking the cofibrantly generated model structure?
Essentially, I am in search of statements of the form "It suffices to show _____ for a generating cofibration." Some such examples can be found here in Propositions 1.4 and 5.2. However, I happened upon this paper after searching a rather wide range of literature, and could not help but wonder if there are more clear examples that come to mind.