An App Store is not an app

For something that is so crucial to iOS, and from a company that generally appears to get User Experience, I'm amazed that Apple haven't done something about the App Store App.

There are some of my issues:

1. Whenever you choose to install or update an app, it takes you out of the store to show you - the icon. Which you can't use. I tend to install apps in batches and check them out later, so this behaviour is completely wrong. 
2. I can't tell that there is a newer version of an app, by looking at the app itself. I have to periodically go back to the store.
3. I'd also like to be able to work through the update list, selectively choosing updates to apply, without having to navigate back to the App Store from where it decides to drop me.
4. The content in the store is static. I'm sure a very high percentage of serious publishers produce a trailer for their app. Would it be so hard for us to be able to see those when it matters - while making the purchasing decision?
5. There's no social context. There are over half a million apps. A (very) basic search model is not going to cut it. This is a circumstance when social search would be incredibly useful. If I know my friends are using an app, not only is it more likely I will enjoy it, but for so many apps, not knowing any other users kills it entirely.

The conclusion I've reached - the App store is not an app. It's features and functions needs to be broadened and improved, and then blended into the overall experience of the OS. The device is the store.

I want to group my apps in different ways, and change the views. What apps have I installed this week for example?

I want all my games to be together, and based on what I and my friends are playing, I want to see recommendations for new games, with trailers.

I want to see that there is an update for an app as I decide to use it, and choose to update to the new version now, later or never.

I want to subscribe to app channels, so I can offload some of the selection workload to trusted curators.

These are just a few ideas which barely scratch the surface of what could be possible, once you forget about the store as an app, and starting thinking about it as an integral part of the whole device.

Any other ideas?