I instinctively strongly disagree with this line of argumentation, but for some time couldn’t figure out why exactly.
Having though about this for a while I think the gist is the following:
(u32, +) is a semigroup, so is string concatenation (String, +). As a result supporting Add on strings makes some sense. Some (many?) algorithms generic over Add will actually work on strings, and provide a sensible result (cf. your own sum example. If I’m not mistaken a slightly adapted implementation of sum could actually get away without requiring Zero).
However, (u32, /) is not a semigroup. Algorithms generic over Div are unlikely to ever yield a sensible result applied to a Path. On the other hand joining Paths is a semigroup, which behaves almost identically to string concatenation.
As a result of that line of though, you might actually be able to convince me that Paths should have an operator overload for joining them, but that operator would have to be +, so it gets the same “benefits” that String does.