This is a great idea for a feature, I totally support it. A lot more developers are making development groups for their games so this feature will definitely help developers manage their group games.
As you said, currently the only way to get hold of someone in the group for maybe a bug report is by private messaging them. Most of the time the user you message in the group never actually reads the message you sent because their inbox is always full. This feature will solve that issue because multiple people will be able to read the message and acknowledge it. This feature will also allow developers to listen to their player base a lot more easier.
To add to this feature request. Developers could be given the option to add messaging categories like Bug Reports, Features or game support, these could be shown in the form of a drop down list on the send message page. Then the group inbox could be split into the categories the developer chose for the drop down list. This will help make the inbox more manageable because you are splitting the messages up into categories as some groups could potentiality get loads of messages.