I have started to do the same for my current team, taking a lvl 80 subbing into the team once he has discovered the BRC entrance and then doing the random dungeon (non-heroic). This makes it 100% BRC on the random.
As for the "Random" part of the LFD tool, I think it is random with the caveat that they do not have an infinite number of dungeon available and so once the popular dungeons are capped they start to throw up the unpopular ones.
So their code probably is an algorithm which states "check for dungeons with free slots, select based on random algorithm" , this would then drop all the full "popular" dungeons before picking the random dungeon and so obviously a lot of the "unpopular" dungeons would keep showing up when queuing in LFD.