My solution is to start Clockwork Empires from a desktop shortcut that links to the .exe in the install directory, instead. Steam doesn't log my play time or show my Steam friends what I'm playing when I do that, though.
This doesn't happen if you never exit or stop playing the game In all seriousness, we know about the issue, but it seems like we need to contact Valve or change how the launcher works completely. It's just not high on our priority list right now because the popup goes away after a few seconds.
I also find that annoying, but you can still click the button even with the steam overlay on top of it.
Not a terrible idea, actually. I put the "play game" button there because it's the naturally place for the eyes to rest at the conclusion of viewing a document. You know, left to right, top to bottom makes the bottom right a natural resting place. Valve -and Paradox - apparently had the same thought: this very error confounded me playing Crusader Kings 2 for the first time - I sat there for 5 minutes trying to find any kind of start button before giving up on trying to play the game for the day. The next evening, sat down to try again and *finally* found it under the stupid Steam tab.
As Spark says, you can still click it. I realise it's not a real complaint though, I did the exact same thing when it happened, whizzed my mouse around thinking 'grrr' but then realised you could click anyway. D'oh!