I agree Ernst....This is the same behavior that I see now.
I was really disapointed to see this situation re-appear because up until one of the recent updates to Mavericks or iOS, everything was working beautifully for me using iCloud mail across my Mac, iPhone and iPad. Everything was updating properly all the time, badge counts were always correct, etc. I was happy to have been free of Gmail.
I'm not sure what broke it.....but there was an update to Mavericks Mail application and there have been several updates to iOS 7 in the past weeks. Somewhere along the line of those updates it stopped working and now I have the same old crummy iCloud mail badge situation again.....and what's worse is, as you noted, the emails themselves are not updating until you go into the app. So the situation has actually gotten worse than just the badges not updating.
It's a horrible day-to-day scenario to have this problem. Incredibly inconvenient and a lot of wasted clicks, and it makes the Apple services look really silly. If it's not fixed quickly then I will probably switch my forwarding around once again and go back to using Gmail until it's fixed again.