I think you already understand that they are unreasonable people. One answer is that the person who implemented/validated it and the person who decided it needs to be TCP were not communicating well.
It obviously works well enough as is, with the current system, so they probably aren't looking into how/if it works much.
I would just do it whatever way you want as long as you can meet whatever needs they have. Clients often specify that things need to done a certain very particular way, then never really care if it is done differently as long as it works.
I have been told before that a service only uses TCP when in fact when I tried connecting to it, it turned out to be only UDP. They didn't know, and bringing it up to them just made them confused and angry.