Mandated etiquette isn’t good manners. It’s nothing. Things that are forced don’t count
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The problem with that is the statement "good game" would become mandatory, causing it to lose any real meaning. Having it as an option allows people to either
1. Say good game after enjoying their game
2. Not say good game because they didn't enjoy the game

Mandate etiquette?
All one needs do is observe who is the poster boy to know how far that notion gets.
Online, I don't normally say anything after a game, but I will respond "Thanks, you too" or something like that if someone else says "good game". If someone says something stupid, like accusing me of cheating or insulting me for daring to win a lost game, they get immortalized on my profile, but I haven't had that happen here, and rarely elsewhere.
Over the board, at different times I've said either "good game" (and had the occasional opponent say "not really") or "thank you for the game" (and had the occasional opponent say "yeah, I gave it to you"). People will occasionally be bad losers no matter what you do.

Instead of gg try, "Thanks for the game" . . .
If you win or lose . . .
What's wrong with all you guys? . . .
I tried it and was repeatedly reported for harassing my losing opponents. Chess.com members cured me of saying anything. I used to always say “Thank you for the game” when I would win, and I would say “Good Game” when I lost. It caused way too much flack.
Assuming you're not being sarcastic with this post, are you saying that you were flagged as harassing other players for saying 'good game' etc when you won, and your solution to that is to make it mandatory for everyone to say it?
Maybe you should just reserve the use of 'good game' for games that deserve it or games when you're playing someone across the board.

I grew up in an era where the losers always were mandated to courteously acknowledge their opponents victory, by the coaches, instructors, parents, and the community as a whole. It was expected.
I was similarly raised to be polite. I wish there were a way to let someone know you thought it was a good game without opening up a chat. Please consider that the *other person* may also not want to talk. Privacy online is about comfort level. It shouldn't be a requirement to be uncomfortable to be here.

why should I say "good game", if the biggest loser of all time the incumbent POTUS does not
say "good game" to old Joe Biden? If he finally does, I'll change my attitude :-)

Here, you could send a message afterwards.
While Nigel is showing Marty his Marshall guitar amplifiers, he points out a selection whose control knobs all have a highest setting of eleven, unlike standard amplifiers whose volume settings are typically numbered from 0 to 10. Believing that this numbering increases the highest volume of the amp, he explains "It's one louder, isn't it?" When Marty asks why the ten setting is not simply set to be louder, Nigel hesitates before responding blankly again "These go to eleven." (Wikipedia)
That works for you but not necessarily for others. Some people wish to express themselves accurately rather then generally. If I wish to thank my opponent for a game that wasnt a good game then I can simply say "Thanks for the Game" rather than good game. This may sound like splitting hairs but there is an important point. Manners, etiquette, courtesy etc. should be sincere not habitual or mandated, it should also be freely given. To require people to express gratitude in specific ways cheapens it or blunts its affect. Personally, if someone does not mean it then I prefer not to hear it and I bear them no ill will.

"Why don't people say gg?"
"Because they don't want to chat."
"But what about saying gg?"
"Because people don't want to chat."
"Yes, but it's etiquette."
"It's an online privacy issue."
"But it only takes a second."
"Doing something you don't want to do may only take a second, but it's still something you don't want to do."
"Then send a message."

No big deal, they can just ignore it. I never say anything in the chat btw, only respond. Unless they are trying to distract me.

Why should only the loser say it? Also, why does it matter? I am usually one to say good game, but I don't ever hear it back. I feel like people either hit rematch or they quickly select new game.
It is easy for the winner to say Good Game. It is not so easy for a loser to do the same. It is, however, something everyone can learn to do graciously.
A winner telling a loser Good Game, it just seems backwards to me.
That's a fair point, I just feel expecting the loser to say it first is condescending. It should go both ways if both players are conscious of sportsmanship.
Saying good game is easy for the winner and loser. It really isn't' that hard to utter or type a simple phrase.