Nor automatically or manually I can add Starcraft II to my library. That's quite awful, once is the game I most play.
What should I do?!
We'll continue to help you with your issue in a ticket I created for it. Ticket #11184 for your reference. I will also be locking this forum topic to prevent answers to questions in various areas; that way we can just stick to the ticket. Thanks for letting us know about this issue!