I have been doing some research on the question of vayechal at mincha and have hit a roadblock. So far, here are the questions I have considered and feel ok with as "answered":
why lain at mincha on a fast day? Got it -- mincha is an important time for tshuva when fasting
why lain vayechal on a fast day -- got it (the above article and my own research) -- even though the gemara posits other readings, the sense is that reading which reflects the midos etc. is appropriate
But here's the remaining question -- why lain the same thing in shacharis and mincha? There are other options for readings according to the gemara, and the notion of reading the same thing is unprecedented in practice. In fact, on tisha b'av we DO read 2 different things. Unless one says that the primary reading on every other fast day is the one at mincha (as is hinted to at the end of the linked page) and it was moved to shacharis for some sociological reason with no halachic basis and therefore no change is made in the reading, I can't understand why we don't read 2 different lainings. Any insight would be appreciated.