The latter is very important because in your example it would not really be hidden. If your function does not have the "exn" effect, it cannot call functions that throw exceptions, full stop. Same with any other effect including IO if you want.
Basically function coloring taken to the extreme. In a statically typed language with statically typed effects you actually cannot get surprised, which was your major complaint.
Type systems that support algebraic effects also typically support row polymorphic effects (fancy generics) so you can make a function generic over "color", avoiding the "function coloring" problem.
Now, having said that, why did I say I agree with you? Well because algebraic effects are a lousy user-facing feature. You almost never want to implement your own handlers, at best you'll plug in a custom handler for the IO effect and that's about it. (And for exceptions of course, but that's just exceptions with extra steps)
Where they shine is for the language implementer. They provide a framework on which exceptions, generators, async/await and even prolog-like backtracking can be implemented, while (very importantly) defining how they compose. That's really the bit that makes them so interesting from a research point of view and why they might make it into the mainstream languages, even if the language doesn't actually ever expose them for you to use.