integration - How is it that treating Leibniz notation as a fraction is fundamentally incorrect but at the same time useful?

I have long struggled with the idea of Leibniz notation and the way it is used, especially in integration.

These threads discuss why treating Leibniz notation as a fraction and cancelling differentials is incorrect, but also go on to say that the notation is suggestive and we use it because it simplifies things:

What is the practical difference between a differential and a derivative?

If dy/dt * dt doesn't cancel, then what do you call it?

In them they say to treat the differential at the end of an integration expression as a "right parenthesis". This throws me off a bit because we can so easily do something like:

$\int cos(3x) \, dx$

$u=3x$

$du=3dx$

$\frac{1}{3}du=dx$

and then proceed to integrate:

$\frac{1}{3}\int cos(u) \, du$

and arrive at the correct answer with "incorrect" notation. I am supposed to treat the differential as a parenthesis but using this notation the differential seems to have a value.

How does this incorrect notation do such a good job ensuring that we do not disobey the "reverse chain rule" and ensures that our integrand is in the form $f'(g(x))\,g'(x)$ ?

People often say that it is very suggestive and I am wondering how. Excuse the LaTeX if it looks weird. This is my first time using it.