So I'm still reading that people have having trouble launching the game through the launcher? I know a solution that might work for some people.

Go find your skyrim folder and create a new txt file named "steam_appid", and within it containing the number 72850. Now save the file inside your Skyrim root directory.

Inside your skyrim folder you'll see a file called "TESV" or "TESV.exe" depending on your Windows settings, anyway make a shortcut to that file and place it on your desktop. Use that to launch instead.

With steam_appid.txt containing Skyrim's AppID you should now be able to directly launch the game's main EXE to play the game instead of going through the launcher.