Allow all games. Remove the prohibited games list.
The Twitch community is composed of people from around the world, coming from unique political backgrounds and cultures.
Prohibiting games enforces a specific view of the world to the Twitch community as to what is acceptable or not, censoring content with valuable gameplay and cultural content in the process.
As of this writing, the list of prohibited games are either due to violence, sex, or any game with the "Adults Only (AO)" ESRB rating. While this could be argued to be a good thing for kids, Twitch has already a "Mature Content" flag in its channel settings for this specific case.
Responsible adults however are not allowed to take the matter into their own hands and are forced to stream and listen to content that do not fully match their own values. This is in my opinion unacceptable since it hinders the creativity of the video game art form and its economics.
This is not just limited to the entertainment factor of violent and sexual content, but also derivative content that could not exist without those elements. For example, Manhunt 2 contains in my opinion a very strong anti-violence message in its endings depending on the player's actions throughout the game, but said message does not make much sense without the censored content. Similarly, HuniePop can in my opinion empower shy people to reach out for real life relationships through the tips given by the characters and by gamifying their fear, but would have a lot less impact to them without the naughty rewards.
Regardless of whether you agree with these opinions or not, the point is that even the most insipid-looking game can have a positive effect to various people, and blocking such games would prevent people from benefiting from them.
Please allow responsible adults on Twitch to enjoy the content that they want freely.

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MonkeysnUncle commented
Something to beware of with a suggestion like this are global Internet laws. Basically, any content viewed online MUST meet the laws of BOTH a viewer's local laws AND the laws of where the provider's servers are.
Twitch is based in America, which has much stricter laws on certain content than other countries and other countries can have even stricter or more relaxed laws than the US in other regards. This isn't a matter of age verification. One of the games on the banned list is a game which features content that is illegal in the US (****, non-consensual underage sex). This game is legal in Japan, but not in the US. Someone watching a video of this game in the US could be arrested and fined, which would also make Twitch legally liable if they allowed viewing of it on their platform. Personal opinions on the content do not matter because the law doesn't care about the accused's personal opinion on the matter. Yes, Twitch needs to reconsider what games are banned and what games are not, but it is a good thing for them to have a banned games listing.
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Shadowed_lady commented
Unless Twitch comes up with a valid age verification process (which I would LOVE) we adults will sadly have to watch what we do around the kids they let on the site.
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MrNY2Cali commented
Twitch allows users age 13 and up. The mature content flag isn't really intended for "age" related things its meant for Ads that might play. Twitch does 0 age check verifications. The no adult content is also a federal & international law thing. Twitch doesn't want to answer to federal or EU regulators. They are NEVER going to allow Adult games to be played because of this.