Do you think there are no reputable casinos where KYC is not required?
One of the most important and fundamental features of cryptocurrencies is anonymity. In my opinion, cryptocurrency casinos should not require KYC verification. Of course, if you prefer to undergo verification, please, you can do it anytime and anywhere, but in my opinion, if someone plays cryptocurrencies and does not want to undergo verification, should have such a choice.
And there are reputable casinos which kyc are not mandatory, some will automatically ask you for kyc when you exceed on their given free limit.
In most casinos there is no information that there is a limit. Usually, it is written in the rules that they can ask you for KYC verification at any time.
How does he know that a reputable site will take care of his kyc? Did he try working on that specific field already? Facebook is also reputable but there are reports that Facebook sells the data of its users.
We don't know maybe the same thing are already happening in the background when we submit our kyc on our favorite crypto gambling sites. We have no idea because no one have ever exposed it yet. It's not only about if our kyc is safe or not but it is also about privacy. Some prefers to be anonymous.
This is a separate problem. Personal data is a completely normal product these days. Whole databases are available on the Internet, sometimes even for free. We will never be sure that our casino details are safe. This is another argument for looking for an anonymous casino just like OP is doing.