Any exchange that requires you to KYC, it means they care about the origin of your money. If they don't care about the origin of your money, are you a criminal? Why do they force you to KYC? It is true that exchanges need users to build reputation and revenue, but first, they need to survive and to do that, they need to comply with government regulations. That is why they always ask us to verify our identity and comply with their terms before using their services.
As for handling forfeited assets, I think everything is clearly stated in their regulations, it's just that most of us don't read them before signing up

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In my opinion, about any exchange require KYC because following the government regulation or the exchange operation close totally if allow their user without KYC for trading, deposit and withdrawing fund. Recently many exchange before have limited amount for withdrawing or depositing before applying KYC but right now most of the exchange have change the regulation if want trade must pass KYC firstly.
For exchange have regulation about where are the money come from and verifying document ID firstly make easily for exchange detecting our deposit come from, the exchange regulation about KYC make us trade, deposit or withdraw fund without problem and the exchange have follow all government regulation.
I think all centralized exchanges have mandatory KYC and this has been going on since years ago. They will always care where the money is coming from but in my mind, the chances of getting flagged are a lot higher when you are talking about bigger amounts (except for maybe Binance based on the comments here). Similar to when you would like to send money from the exchange to a bank account. They will monitor it but you will never really have issues if you stay below certain numbers (<3000 €). At least that is valid for my country.