I'd go stronger than this and say that Leibniz's notation is actively harmful. It is very useful for quickly doing certain kinds of computations, but at the expense of conceptual understanding for students. Obviously, it's fine to use whatever computational aids you want when you understanding things, but most students are taught nothing but this fragile notation.
It's useful for solving (or partially solving for a set of conditions) linear differential equations, say in the fields of mechanics or electromagnetism. One can work with dx and dt as if they were just factors and move them around quite intuitively.
In the same line, it's great at shining a light on the substitution rule for integration.
Given that your point that it can be obscure at first remains valid, I'd walk the middle line of introducing students to the f'(x) notation first; and after the introduction to integrals introduce this notation to them.
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.
