Since the rearranging part of your strategy involves dividing $dy$ by $dx$ this post on MO is quite related to your concern.

As far as the arc length is concerned, I usually avoid your strategy as a teaching tool at the outset, but sometimes mention it as a way to memorize the formula. Why do I avoid this? I have no conceptual way to distinguish between $dS$ (the infinitesimal arc length) and $dx$. They are infinitesimally small so "intuitively" they have to have the same length; in particular $dy$ is also infinitesimally small give additional support to the intuition: In a right triangle with such an small side $dx$ is equal to the hypotenuse ($dS$). Right? Indeed, once as a student, I did something similar to this line of argument when calculating the area of a surface of revolution for the first time on my own.

Since the rearranging part of your strategy involves dividing $dy$ by $dx$ this post on MO is quite related to your concern.

As far as the arc length is concerned, I usually avoid your strategy as a teaching tool at the outset, but sometimes mention it as a way to memorize the formula. Why do I avoid this? I have no conceptual way to distinguish between $dS$ (the infinitesimal arc length) and $dx$. They are infinitesimally small so "intuitively" they have to have the same length; in particular $dy$ is also infinitesimally small give additional support to the intuition: In a right triangle with such an small side $dx$ is equal to the hypotenuse ($dS$). Right? Indeed, once as a student, I did something similar to this line of argument when calculating the area of a surface of revolution for the first time on my own.