I did some consulting work on a broadcast video server once. I don’t remember the in-depth details, but the video was stored on the server in a proprietary format based on M-JPEG. The master control would queue up the segments which would then be transcoded for analog broadcast (I am less familiar with this part of the process). There was a lot of custom FPGA based hardware to make this happen very quickly, and the server itself was running VxWorks (a “real-time” operating system).
I don’t think many stations use this product anymore however so I doubt this is the norm.