Not doing KYC on the black market doesn't mean it's more safe and maybe it could be more dangerous because black market sites usually are planted with phising or malware that could damage our PC or even could access our PC if we are not careful clicking anything on the websites and they could steal our wallet.
I will definitely do KYC in a big and trustful exchange, there are a lot of good exchange with exposed CEO and teams, so if we know the owner of the exchange why it's so bad if the owner know the user ? i think it's normal to know each other in any business, except we are doing a money laundrying or doing anything wrong getting the cryptocurrency.