AC21 is safe if you have a copy of your LC with you and ensure that the new job description is almost the same as the one mentioned in your LC. For my personal case, i have never used AC21 because i thought that getting such an ideal offer (with same job description as in LC) will be difficult. We use AC21 to move to bigger companies and i wasn't sure that they will cooperate with us for giving the document of our choice. Secondly i was foolishly over optimistic for a large part of time, always assuming that i will get approval very shortly. Only after i got my GC, i felt that i have unnecessarily waited too long without changing jobs.
Coming to changing jobs after the GC, i would say that just wait for 2 or 3 months and then change. I wouldn't care for the cautious trash thrown by some lawyers. I myself changed job 3 months after getting my GC.
To answer the main question of this thread, i would say that changing job soon after getting GC is much safer than using AC21. I thought of changing job 1 month after getting the GC, but it took time for me to get the job of my choice.