You are correct, most gamblers don't feel satisfied giving their details out, they feel it is not safe for them. I think if casinos takes kyc very strict many gamblers may lost interest to gamble with casino which they wouldn't feel okay giving identification out. Gambling companies feels Gambling being private will be better for customers. The truth is that even if casinos begins kyc underage people will still look for a way to gamble. Kyc doesn't mean there won't be any way or any form of gamble for underage people to participate in gambling.
Strictness in KYC can limit underage gambling for a while before they get very creative and find ways to bypass the checks either with identity cards of their parents/ guardians or most likely carefree grannies. Technology is getting mad and these dudes are getting smarter. for me the only way to checkmate teenage gambling seriously is to enlighten parents to install, activate and intensify on parental control features in teenagers' gadgets, that way they are updated with their activities and can provide that level of guidance locally.
I am one person who detests giving out my details and I surely prefer maintaining anonymity in my gambling activities.