On iPhone (at least; I don’t have any maxed-but-not-championed characters on other devices) you cannot champion a character from the “Train” screen. If you click “Train” and select your maxed character, the “Promote” button does not appear.
Instead, you need to go to “Roster” and select the character from there in order to champion them.
This is needlessly confusing 
It is there on iphone, but you’re not doing it right. After clicking Train and choosing your max level character, click on the “Raise Level” button- then you can champion with no problem.
That being said, it would be helpful if the “train” button were replaced with a “Promote” button for characters at max level that have not yet been championed. I guess that makes this more of a suggestion than a bug.
It says train because you can still train cover abilities at max level to change which covers are maxed.
Sure, but after you promote them you can do that as often as you like. So that’s not necessarily a helpful option anymore.