The case can be made that he should have shut down the engine, after all he did get killed. But, that doesn't mean he SHOULD have shut down the engine. If there was somone else in the plane, then not shutting down the engine isn't neccesarily the lapse of judgement that you might think...especially if he was on a taxiway. It's quite common in GA to get in and out of a plane with the engine running if there's a qualified someone guarding the brakes.
Flight Instructors do it all the time with first solo students. After doing a couple of warm up TO's and LDG's, they'll taxi to a ramp area and the instructor will get out and let the first solo student taxi back out and do their 3 bounces. Seat swaps are comon too, when you have two students aboard, and they're taking turns observing each other train.
If he was alone in the plane, then he should have shut it down, no question.
Obviously, this well seasoned FI made an incredibly unfortunate boo boo, and that's really to bad.