In the case of units which have almost finished with training, I think the devs we're just providing the info in more than one place to be sure you didn't miss it. You should definitely continue to read through the event log in its entirety each turn, line by line. It serves as a good memory jog for me. It's easy for me to forget what I was doing with some obscure stack of units in a distant corner of the map, especially if I have a fresh set of problems at the beginning of the next turn; so, I find the movement logs, for instance, very useful.
For some information, like the arrival of new generals or the promo ability of other generals, there is no other in-game notification of which I am aware. You might miss something important, which can affect your strategy, unless you happen to discover it later by chance.