I hope you will forgive me for this wall of text, but this is a topic that is quite close to my heart and that I've gone back and forth in many times before settling on my current perspective.
I appreciate that especially for mathematicians and programmers, making a clean distinction between a function and its evaluation is a key conceptual point, and Leibniz notation obscures this fact. However, there are good reasons why physicists use Leibniz notation, and this answer really glosses over that.
The reason is that the particularities of the mathematical structures used to model a physical problem matter a lot less than the relationships between the actual physical underlying quantities. And there can be a lot of them. The answer evokes thermodynamics, and I couldn't think of a better example. Are we really to introduce a distinct symbol for _every_ possible functional relation between _every_ state variable? If I have T = f(P, V) and P = g(T, S), do I need to remember in which position exactly each function has been ordered before I can write down "how P varies with T when S is fixed"?
Leibniz notation, although formally tricksy, is just the best tool for communicating the _intent_ of a physical relationship without getting bogged down in the mathematically detail. The purpose of an equation is to express that relationship to the reader. Think if it like code — is it so bad if my code does a bit of magic behind the scenes to allow for a clearer reading, even if the semantics aren't immediately obvious? Well, ultimately, it depends on the situation, and a balance must be reached. I don't believe that expressing everything the way TFA suggests is striking the right balance.
Notational abuse happens all the time in physics, and this is certainly not the most egregious example. Just compare it to the path integral. It's easy to assume that this is because of a lack of sophistication or rigour by physicists. (Certainly I did throughout my physics education, being more mathematically or pedantically inclined.) But it's a simplistic view.
Now, while I'll defend the usage of these sort of unrigorous conventions even if they are strictly speaking meaningless mathematically, what I won't defend is the slapdash approach that is often used to _teach_ partial derivatives to physicists. Some exposure to concepts like distinguishing real variables/quantities from functions is needed, or, as TFA does mention at the end, the student won't be able to unpack the notational convenience into clear semantics, which can lead to unclear reasoning. I used to share the views of the author for a period when first introduced to Leibniz partial derivative notation in my first thermodynamics course, and, probably like them, found it to be totally incomprehensible symbol soup. But for myself, I see now that it was mostly a failure of teaching rather than a failure of the notation itself.
I'll add one last thought. There is a degree of "primitive obsession" at work here, trying to fit everything into positional functions and real numbers. I have thought that a formalism that better reflects the structure of "physical quantity" (as opposed to thinking of them as plain real numbers) may help bridge the gap between rigour and conceptual convenience. The tools are really already there. We need two key concepts: first, borrow from programming the idea of keyword arguments (there is a book which sadly I can't remember the name of which does as much to formalize Einstein notation for tensors in a coordinate-free way); second, to model quantities not as real numbers but as differentiable homomorphisms from a state space (modeled as a manifold in a coordinate-free way) to the reals. This is how physicists already think about it, it needs only be formalised.