...If anything in iOS really needs an update it is the homescreen. There isn't even a visual indication that an app is running in the background, using GPS signal or downloading a file...
Well there are indications some of those things are happening:
App running: "running" is loose term on iOS. An app is obviously running when you have in on screen so no need for an indicator, but is it "running" when it is not consuming any CPU time but is still loaded in to RAM (suspend) as most apps are when not on screen? Are apps "running" when they're still shown in the recent apps list but have been swapped out to storage and aren't even using an RAM? When in that chain of events does the "running" indicator go away? (Hint, the answer to both questions is "no").
GPS signal: The indicator is on your status bar. One or more apps are using location services when that arrow is displayed; Settings>Privacy>Location Services to see what arrows are blue and tell what is using, or has used in the past 24 hours, location data.
Downloading a file: Again a single status bar icon of rotating arrows indicates network download activity by one or more apps.
To forgo the simple status bar indicators and put the indicators on each active app icon leads to these questions which need serious pondering and design:
1. Do we invent new indicators like badges, change the icon shape, color or alter current badges to apply these status indicators?
2. Do we want to scroll through all our home pages to identify what apps are using these services or are we better off with a single list in settings?
3. Do status indicators show though folders or do you need to open each folder to check for status icons?
4. Is there some gesture or display that should consolidate all of this information that 95% of users don't care about to keep things simple?