Math is at its heart about intuition; formalism is a just tool. One uses the formalism but one has to see past it.
Roughly its the difference between knowing that a group is a set with a certain binary function, and knowing that a group is a particular formalization of symmetry.
While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.
To me this would be another way of developing intuition, via analogy across disciplines. E.g. Volk's Metapatterns draws patterns (but perhaps less rigorous and more loose) across discplines giving one intuition about many things.
In this case, seeing the connections across the disciplines isn't necessarily something I see or am pursuing because I'm searching for a shortcut (although I did mention O(1)), I just see that as a side benefit. What appeals to me more is developing the intuition you mention via the analogies. I feel like if the idea is restated in different ways (not necessarily just terms, but perhaps visualizations/re-conceptualizations) a more holistic/intuitive/fuzzy idea emerges rather than anything rigorous necessarily, but you know a lot more math than me so maybe I'm saying nonsense.
To be honest if you're sufficiently motivated and you find it interesting then go with it.
Better to be motivated about an idiosyncratic method than ambivalent about the whole thing.
While a shortcut to understanding sounds nice, I don't think you'll find it by focusing on abstract formalism.