There have been some great suggestions to allow flexibility around configuring the organization landing page, and even a custom landing page for individual users.
Something else that would be of value is to have a "Page Item" inside the organization. The idea would be virtually the same as capabilities already in place with the Open Data platform for ArcGIS Online. However, there's a limitation that "pages" in ArcGIS Open Data can only be shared with the organization, or with everyone. There's no way to share a "page" in Open Data with just a group, or with invited users within a group.
The use case here would be to allow an organization to have multiple, configurable, landing pages. While it's true there is the group content page, and the gallery app, there's a need to make a "home" for groups where there can be HTML text, images, media, embedded apps (again, look no further than the pages functionality of Open Data). Office 365 offers similar capabilities with Group pages. An Organization could have a departmental "page" for Public works. This is a bit more than the current group content her have now (either as the current group page or a gallery). On my departmental page I could not only have the list of data sets, but I could embed a dashboard element or chart, post announcements (as HTML text), and search group content. Yes, some of this is here in the current group pages, but there's a need for something with a more interesting user experience.
While a "page item" inside of ArcGIS Online (and Enterprise Portal) would solve the problem, another solution would be to modify the sharing capabilities of Open Data sites and pages. Right now, as mentioned, Open Data sites and pages can only be shared with the "Team", the "Organization" or "Everyone". No group sharing.