If you want to generalize the intuition from "minimal cell structures" in topological spaces to model categories, you actually probably want to be thinking about cellular model categories rather than combinatorial ones. I think of combinatorial model categories as a good place to generalize arguments based on powerful results in category theory, like results that work well for simplicial sets. But, the category of topological spaces is not locally presentable, so arguments of combinatorial model categories don't work. But, the notion of a cellular model category was invented with topological spaces in mind.

The best source to read about cellular model categories is Hirschhorn's book, specifically chapters 11 and 12. Chapter 11 gives a whole theory of cell structures (known as cell complexes) and Chapter 12 explains how in a cellular model category, that structure lets you do homotopy theory. There's a theory of subcomplexes, intersections of subcomplexes, and bounding the size of a cellular presentation (section 12.5). The main motivation for cellular model categories is to do Bousfield localization, and section 4.5 explains the Bousfield-Smith cardinality argument, which is done first for spaces (in chapter 2) and then for cellular model categories in 4.5. Definition 4.5.3, Lemma 4.5.4, and Prop 4.5.5 are all about finding small cell structures, small enough to make the cardinality arguments work. The material in Section 10.6, about cellular presentations and subpresentations (smaller ones). The theory stops short of trying to get a *minimal* cell structure, but it does set up a poset of structures. So, if you had a model category that was both cellular (to have cells) and combinatorial, then you could probably get a minimal one. This kind of question reminds me of left-determined model structures, for some reason. But, since Hirschhorn tells you how to take intersections, maybe you don't even need local presentability.

Now let's think about your two examples. First, models for spectra (build on either spaces or simplicial sets) are cellular, so rational spectra are too. Model structures on chain complexes are also commonly cellular (and combinatorial) if they are cofibrantly generated. So, you certainly have a "sphere-disk" cellular presentation like you want, and can try to generate a minimal one via the intersection theory of cellular presentations from Hirschhorn's book.