This is exactly what I remember from every physics course I took in my life: many physicists use math as a blunt tool rarely even bothering explaining them, and

This is exactly what I remember from every physics course I took in my life: many physicists use math as a blunt tool rarely even bothering explaining them, and almost never justifying why the tool is actually applicable to the problem at hand (e.g. by simply stating that the axioms of the math seem to align with physical observations at all times, thereby very much justifying the use of the mathematical framework). Far worse, many a physic teacher doesn't even understand the tools they're using from math: they're just parroting and regurgitating from memory a combo of the textbook and what they've been told all the way to grad school, and if you happen to manage to corner them after class to dig a little, you quickly discover how shallow their understanding of the math actually is. Both Thermodynamics and Electromagnetism were initially hell on earth for me for this very reason: the profs were both incapable of explaining what the tools were and did, how they worked and why they were applicable. All they wanted was for you to absorb and memorize the formulas and stop asking all these stupid questions about the math. When I finally understood the math tools underlying the two subjects, both thermo and electromagnetism became crystal clear, but boy was the initial exposure an effing headache. A very good example of this is the famous book "div, grad, curl and all that" that many people hail as excellent. Well, I strongly disagree. It was written by a physicist, and the whole book is laden with references to either fluid mechanics or electromagnetism. It never is capable of explain things without making reference to physics when the tools are in fact completely independent of it.