And even in this case, there is a non-zero probability of theft of personal data from the casino that you have chosen and consider reliable. The reasons may be both outside this casino - these are hackers who break into databases. And inside the casino itself - these are unreliable hired workers who steal this confidential data to take revenge on the casino or simply out of hooligan motives. As a result of these two factors, the likelihood of leakage of personal data increases even in the most reliable casinos. But if you have not passed KYC and the casino simply does not have information about you as a person, then such thefts are definitely not scary for you and cannot affect your life in any way as another source of problems that lie, of course, in the illegal sphere. I mean embezzlement, extortion and other possible criminal manifestations.
That case will be there as we know that there is no 100% safety when we are online or sending our document to some parties. We even don't know if inside casino will be a place for some people who want to gets the document for their own interest. We must really be careful if we want to send or verify ourselves to pass KYC and make sure that the casino have its reputation and popular so we will not have to worry with anything. They needs license to have their reputations among the other casino and that can make their members believe with them and willing to do KYC with them. We really don't know but we can verify before we continue the process to do KYC and we have a choice to still use the casino or moves to the other casino.
However, no amount of checking the license of a given casino can help in the question of how well the personal data obtained during your KYC verification is securely stored in this casino. Simply because it is the licensed and well-known large and famous casinos that are most often subjected to hacker attacks, some of which are successful. And it is from such casinos that corrupt employees with access to databases steal these very databases.
So I would not consider large famous casinos more reliable in terms of storing your personal data.