I absolutely love the combat speed-up option. The auto-battle on the other hand I don't really see the use for it. It doesn't save any time since you have to watch it all anyway. Maybe if there wasn't a speed-up option I'd use it - set it going and go do something else. Plus, I wouldn't dare use it in arena. The only situations where I would feel safe using it, really they should be looking at why we're given such trivial fights in the first place.
What I do like about it is that it's entirely optional and thus not the slightest bit harmful.
What would be nice, by the way, is if it remembered your previous selection for the battle speed so I don't have to set it over and over again every battle. But please, please do not implement such a memory for auto-fight!