Why am I unable to play CS:GO through Steam Family Sharing anymore?

When you mean:

Actually, I mean it shows "purchase" where it should say "play".

and

Ok thank you, but the problem still exists, I said that I can play all other games except CSGO, you said I can't play his "games". So basically I cannot play CSGO when he isn't playing anything, I can play other games.

It is because:

Counter-Strike: Global Offensive has been excluded from Steam's Family sharing services.

It says so right here:

Besides, wasn't fully open anyway when it was accessible through family sharing. You could:

• Not access any official or VAC-secured servers
• Not have an inventory, even with actual CS:GO items in your inventory, and:
• Only access insecure servers or play offline.

Did I mention it's also free-to-play now?