I think there is nothing wrong with reasoning using differentials/infinitesimals in introductory calculus and physics classes provided the nature of the reasoning is made clear and explicit. More precisely, my contention is that in general such uses serve to motivate definitions rather than to hide hard-to-formalize details of proofs, and it is operationally irrelevant whether such motivational uses are formally justifiable.
(I am conflating differential/infinitesimal here because, although in general I would not conflate these notions, in the context of the question that was asked, it seems to me that a differential $dx$ is being treated as something very small but not zero by which it makes sense to divide, so the while the notation is for differentials the sense is for infinitesimal.)
Personally, when I teach these things, I do not write $dx/dy$ to mean divide the very small $dx$ by the very small $dy$. I write instead $\Delta x$ and $\Delta y$ and $\Delta x/\Delta y$ which I speak of as change in $x$ and change in $y$ and treat as small, but nonzero, quantities. I do this because I don't want students to get in the habit of operating too freely with the Leibniz notation, not because I don't like the notation, but because experience suggests that free manipulation of such notation causes a lot of difficulties for students inclined to interpret everything as a series of formal manipulations.
In the specific example of the arc length of a curve, it is important to keep in mind that without calculus one has no definition of the arclength of a general curve. One knows what one means by the length of a line segment (having fixed beforehand a length scale), but extending this even to graphs of polynomials is not obvious (it is exactly the sort of problem that gave rise to calculus).
The naive approach, which turns out to work, is to approximate a given continuous curve (but what does continuous mean without some notion of limit/infinitesimal? - so if you like restrict to the graphs of polynomials or real analytic functions or some such class of curves whose members are de facto continuous but where this does not have to be proved) by a polygonal curve joining some points along the curve that partition the curve into subcurves. It is natural to view the length of this polygonal curve as an approximation to the as of yet undefined length of the curve in question. Doing so one obtains a sum plus some errors terms, and the sum in question can be interpreted as a Riemann sum approximating some so far unformalized integral. Moreover, its form does not depend on the chosen partition (of course its value does). The error terms can be controlled formally using the Taylor approximation, or can be treated as infinitesimals in some hand wavy way - it really doesn't matter - because one is not going to prove that in the limit the errors are negligible and the remaining sum converges. Rather, one supposes the limit exists, or, more properly, calls the curve rectifiable provided the limit of the sums exists, in which case that limit is precisely the integral for which the sums were viewed as approximations. Hence one says the curve is rectifiable if the corresponding integral exists (its precise form is irrelevant to this discussion), and when it does calls its value the length of the curve.
That this is a reasonable definition is not something to be proved beyond the almost trivial observation that applied to the primitive objects used to motivated the definition (that is to line segments) it recovers the primitive notion of length, and in some other essentially physical examples, such as the circumference of a circle, that it yields the known (empirically) answer. Consequently, it matters not at all that some informal notion of infinitesimals was used in motivating the definition. Its correctness (beyond its self-consistency) is a thing judged by empirical and aesthetic standards external to the mathematics.
In general, most uses of infinitesimals in introductory calculus and physics courses follow this paradigm. The purpose of their use is not to provide an informal or heuristic verification of something that could otherwise be proved. Rather, it is to motivate the extension of some primitive concept, such as length, area, work, circulation, etc. to a class of geometric objects for which the primitive notion is by itself inadequate. The correctness of this extension is judged by its utility or mathematical beauty, or whatever other external criteria are appropriate.
One can go astray in this sort of reasoning, and this does happen when it is applied in more complicated physical contexts where there may be more than one natural way to realize the approximation of whatever thing needs to be computed (defined! to compute something one first has to define what it is one computes) and they may lead to different limiting interpretations, with different results. In physical contexts one is correct, one is not, but this is judged comparison with experiment (something like this underlies the discrepancy between the vakonomic versus nonholonomic interpretations of velocity dependent constraints in classical mechanics - one of which is generally incorrect for real physical systems). In purely mathematical contexts one might (if one gets lucky!) obtain two different theories, and, while one might be judged more interesting than the other, it might be hard to call one more correct than the other.
