I originally made my Facebook account in 2007 to play a Flash game called Jet Man, which was basically Flappy Bird before Flappy Bird (I want to say the original famous version of this game both were ripping off was called Helicopter Game though?)
You would invite friends to play the game to unlock more characters, including copyright violating ones such as Super Mario, so I assume the game was removed from Facebook long before the death of Flash.
That game, and some mafia game, was what got our whole class to sign up in 2009 or so. Or, rather, if you didn't sign up someone made an account for you, so they could claim the benefits, hehe.
Why did they drop games? While I hated people forced game shares in my feed as much as the next guy, it seemed like an extremely successful branch of Facebook features.
I don't think they've dropped them, you can still find them under the Gaming tab.
But their heyday was before the mobile wave, after which the two app stores became their main host while Facebook failed to become WeChat (albeit they still brilliantly/rapidly pivoted to mobile-first).
You would invite friends to play the game to unlock more characters, including copyright violating ones such as Super Mario, so I assume the game was removed from Facebook long before the death of Flash.