I strongly disagree about what you have said, because even if it is a trusted sites, you don't know them personally, and you don't even know what they are going to do with your "personal documents" that you have said, it is just about playing safe in this vast internet world by not giving personal information that fast without thinking what might possibly happen.
The worst thing scenario by giving your legit personal information online is that they could use it on something that is not right, like they are going to use that to pretend as someone to scam somebody, or even worst than that.
No repeated site will use your information to scam anyone or use if for anything negative. Most times gambling sites even ask for KYC is for the compliance of their home country and they can assure that your information is in safe hands.
The reason why most of you fear to drop your kyc is because you personally dont trust yourself. If your hands are clean and you have no fear of committing and offense that might make you get tracked down by the KYC you must have submitted somewhere, then you would have no fear on submitting KYC.
Its a thing of choice, if you hate to submit kyc, then gamble on site that do not require for, but dont see those who ask for it as scam.