Most of licensed casino needs KYC at some point that's why if you are not ready to give your personal information it is much better not to play on those sites and risk your money to unregulated gambling sites. I would choose to go on KYC on reputable site rather than risking money on no KYC site that can turn a scam anytime.
Sad thing is, Only few Casino has no KYC thingy on there ToS and the majority of the casino that suited for your description are those without a license or in short not trusted. So far I still didn't play on a casino with license that has no AML policy section because maybe it's the number one requirements for there license. I agree that we should only play on trusted casino so that in case they will ask for KYC. We can guarantee that our credentials is safe.
I would say that no KYC is not a good and secure thing for you neither. It is clear that if you are gambling at a place with no KYC then what are you going to do? This is the trouble, the worst thing you could do right now would be to actually give your KYC to a place that shouldn't take it, so I understand your fears.
However, approach it this way, if you share your KYC with a trustworthy place, then they will use that to protect you. Why not give it willingly, and then you could feel at ease in case something goes wrong, and that way you could use your KYC, your own picture and all, to get back to how things were without worrying about what could happen to your account.