Yea, I agree that, in addition to computational speed, the Leibniz gives some reasonable intuition for some formulas. However, it can also give bad intuition in a multi-variate setting, e.g., dxdyx/dzdw.

A good self-check is to see if you can convert from Leibniz notation to a more rigorous one at any given step in the computation and understand that step rigorously. Personally, I find that functional notation (using D as an operator on the space of functions, etc.) to be as simple to use and much more likely to alert me when I'm about to confuse myself.