Why isn't ffmpeg used in movie production or broadcasting tool chains?
I work as an assistant editor on feature films, and use ffmpeg all the time, primarily for two purposes:


Transcoding files to be uploaded for producers to view on digital dailies systems (Dax, PIX, etc). I've written up shell scripts that accept property-of and recipient strings as command line input, along with target bitrate, and then generate the desired files with a watermark made from those inputs. It's convenient, faster than AME, and more stable than a Compressor cluster. When I figure out how to do SMPTE timecode and feet+frames burn-ins with ffmpeg, my use of the application will expand further. 
Transcoding files for editorial whose original video or audio are in codecs or formats not supported by AME or Compressor. For example, clips needed for temp can be pulled from BDR or DVDR video discs, ingested into an mkv file, and then from there transcoded with ffmpeg to editorial-ready DnxHD or Prores.


I don't use ffmpeg for any sort of final or critical output, and I only use it in a few limited ways that I've extensively tested. But in answer to the original question - I do use ffmpeg a ton on both film and TV work, and think it's a great tool that helps me get through many of my regular tasks quickly and well. I couldn't guess how many other people in features or TV are using it. I frequently recommend ffmpeg to other AE's, but most folk have a command-line aversion.

ffmpeg's use in professional film and broadcast settings is probably also limited by the learning curve of not just the application's own syntax(es) but also the exposed complexity of video formats. Sometimes in ffmpeg a user has to know a ton about source and target codec in order to get things done. Use Compressor or AME to transcode from one codec to another and you'll never have to deal with perusing a long list of different YUV pixel formats, or having to make sure that your target file's height is divisible by 2 to avoid an x264 error. Those of us who enjoy learning arcana and carving our way to solutions are probably silently using ffmpeg at work.