That makes your Newton method require calculating the Hessian of the objective function, which is a second derivative. A normal Newton method for nonlinear systems uses the Jacobian, which is the first derivative (the connection is that the Hessian is the Jacobian of the gradient). But indeed, formulating it so that you have discontinuities (absolute values) and require second derivatives instead of the first for the same algorithm is not the good kind of magic :-/. You could then try using only gradient-based methods, like Adam, but you can see that is actually using less problem structure. BFGS for example is related to Broyden's method and is thus using simple Hessian approximations, so for nonlinear system solving this is effectively less stable than just using Newton.

That's not to say that there's no reason to use this trick though. I have seen that some optimization tools can improve robustness in some cases, and this is a way to make use of heuristic methods (genetic algorithms, differential evolution, etc.) for globalizing the process, but it's probably not the first trick you'd want to do in most cases.

While raising derivative order is one way to understand why this should be less efficient, but another is to understand that it loses some specific structure. In the nonlinear system problem f(u)=0, any u that solves this is a solution. All solutions are the same, if you find one you're good. In the optimization problem, you want to find "the" u that minimizes f(u). Now if you are minimizing f(u)=||g(u)|| and you know there is a u that causes g(u) to be zero, then all u that causes g(u) to be zero is a solution, all match the global optima. But solvers don't necessarily know that. Solvers need to use second order information to know that they have not hit a saddle point or other behavior. They don't know that "near zero = done", and they can't specialize on this information. This is another way in which you're just giving sophisticated more solvers more work to do where you already know the answer by problem design.