Uloli: play button is still greyed out. There is no exe file, it is under the GOG client
Well, there *is* an .exe, you just might not know where it is. I don't use the Galaxy client, so I'm not going to be as much help as I should be. There should be a settings icon, used to be bottom left, don't know if it still is. If you click on it, you'll be able to, among other things, change the install path for your games. If you see what it currently is, you can go to your file manager and find the folder where Galaxy installed Witcher 3, and in that folder you should have a folder called bin inside of which should be a folder called x64, which will be holding the executable, probably still named witcher3.
If you can find that, then you can try running the game that way. If that works, you can make a shortcut to that executable and put the shortcut on your desktop and just click from there to play.
If you want the Galaxy client to be as problem-free as the Steam client, you need to cryogenically freeze yourself and come back in a decade, when GOG has had as much time as Steam had to fix things :P
EDIT: I got lost in my words and forgot to say, if you aren't happy with that as a solution, the only
simple fix I can think of would be to just uninstall Witcher 3 completely, set the client to re-download the game and updates, and go to bed, then play it tomorrow.
EDIT 2: If you have a large hard drive, or a removable one, you might even find it helpful to select the option in Galaxy to back up the installers, when you download a game. If you ever need to re-install a game, you already have the files and don't have to download them a second time.