If you are having trouble getting Log Shipping to work in SQL 2016, make sure you have .NET 3.5 installed. The prerequisites state .NET 4.6, as does the version checker in the installer. However, it seems that the log shipping binary relies on .NET 3.5.
I noticed this when I tried to enable log shipping and the jobs kept failing. The error just said that it failed at step 1, with no further information. I then looked at the command line for the job, pasted it into the command line and up popped a notice about installing .NET 3.5. I installed that and low and behold it started working.
Simple solution to a fairly simple problem, but a bit frustrating until you figure out the cause.