Many traditional calculus classes completely omit all mention of numerical calculation and approximation (in any concrete sense). It is certainly viable to cut Newton's method and it might make sense if students can't properly manipulate logarithms and trigonometric functions.
On the other hand, numerical calculation and approximation introduce lots of interesting and useful ideas (notice I did not say anything about their direct utility). Newton's method is instructive in this sense. Moreover it is one of the few methods accessible at the elementary level that is actually useful and used in practice (Simpson's rule and, later, Runge-Kutta are the other two that come to mind). Teaching it might help to explain how calculus is used for something other than passing exams.
One way of motivating Newton's method is: follow the tangent line until it intersects the horizontal axis. This makes it an instructive application related to derivatives and the tangent line approximation. Often the tangent line approximation (the linear Taylor approximation) is presented as if it were important with little or no application that serves to justify in the student's mind the claimed importance.
Perhaps teaching approximation techniques can help eliminate a bit of the magic thinking associated with calculator use. Students generally have no idea how any digital approximation is actually calculated nor that what devices return are approximations. How in the world did anyone calculate square roots before there were handheld calculators? How do calculators do it now?
It should not be that hard to teach Newton's method in a short, self-contained way. Students typically like algorithms. Both the bisection method and Newton's method are easy to teach and easy to learn. Both are easily presented in a step by step way. It should be easier to teach Newton's method to mediocre uninterested students than it is to teach the chain rule or integration by parts. What can be a bit complicated about teaching Newton's method is that sometimes it fails. However, it may be good for students to see this too.