This is a question I have pondered too. Being a primarily synthetic chemist and one for whom a lot of physical chemistry was never properly placed in a context that made it relevant, I naturally err to the lower rigour, more 'hand-wavy' end of the explanation spectrum, because, as others have pointed out of themselves, I am simply unqualified most of the time to provide much more rigorous explanations.
I think the 'best' format of answer, would be one that is understandable on multiple levels of complexity or abstraction, and doesn't require a firm grasp of the underlying mathematics to follow, but provides them for digestion. This makes the answers more useful to a broader range of readers beyond the level the initial question is pitched at.
This idealised answer I think should touch upon three sections, reflecting the three main levels of comprehension as I see them:
1) Metaphor(s) tying the behaviour to a very well understood or basic principle, for example, in one answer I pointed out that a way of grasping that 2s is lower energy than 2p because of penetration might be to consider Coulombic attraction of point charges. The relevant equation is simple if unfamiliar and crops up in most secondary/high school curricula, and provides an initial handle for the reader to understand the behaviour. Think about trying to explain a phenomenon to a parent or non STEM friend, you aren't aiming for detail, you are aiming for broad strokes, which in this case can be built upon in 2).
Obviously this isn't an explanation and certainly not a rigorous one, but it is supposed to be an 'easy way in'. I gave the particular answer I mention after a number of very technically proficient and rigorous answers that I found very hard to follow even as someone completely familiar and comfortable with the concept of radial distribution functions and most of the relevant QM, where I felt the question was more about being unable to wrap their head around the behaviour intuitively, rather than that not believing the maths.
2) The classical model. Variously hand-wavy, this is the next level of rigour, giving for example details of relevant orbital characteristics in qualitative terms. Most technical diagrams or pictures (particularly orbital diagrams or mechanisms) would come under this section. Here the goal is to explain the answer in a functional and chemical framework which allows common application of the knowledge.
This section is aimed at giving a functional understanding of the behaviour of something, ideally in terms of generally applicable principles. e.g. Orbital hybridisation, mechanism classes. This level of explanation might not be particularly precise, or always a good representation of exact behaviour, but most of the time it serves you well. HOMO/LUMO/SOMO type explanations I would probably consider part of this section as well.
3) The QM perspective. Here is where all the physically minded respondents shine, stepping through logical and formula-based explanations of phenomena, resulting in as quantitative a picture as can be gathered of a given bonding behaviour. This is for the reader who really wants to get to the bottom of what is understood about the phenomena in question.
The one point I will note personally about this section is that often notation choices are made for efficiency or elegance from the perspective of someone knowledgable in the art, not for someone less fluent or only just dipping their feet in. Sometimes it can be easier to follow in plain, bog-standard algebraic notation (and/or with accompanying text detailing the parts of the equation) rather than fancy short-hands with lots of field-specific notation (Bra-Ket and group theory notation spring to mind as obvious barriers to comprehension for the otherwise mathematically competent).
To achieve this total may require multiple distinct answers, but ideally each would at least be considered.