Money laundering mostly stops by wagering requirements. Casino KYC you if they have doubts in your honesty. Unfortunately, such doubts they have if you begin to win. And they have no any doubts when you deposit money.
On one hand - there is nothing to be afraid in KYC. On the other hand - today KYC is used mostly against gamblers, not for its real purpose.
The reason why most person are all very serious about their KYC is because most of the time some casino do sell it out but although some of the casino that do that are the ones with no reputation at all but even at that many gambler are still very conscious of the data they do submit to the casino. But like you said there is actually no need to panic if you are clear beside it's a matter of seconds or minute for the verification of the details you submit for reputable casino.
I understand the challenge with submitting KYC to untrusted platforms but the truth is that the advantages of playing in a casino that request for KYC during registration is higher than those that does not require KYC but might turn around to ask for it when there is heavy winning. That attitude is a sign of irresponsibility wrapped under criminal intentions. There have been countless cases of people losing significant income due to casinos requesting KYC after winning; I consider this a deliberate attempt to steal people's fund.