I suggest not to try KYC with websites that do not have a third party KYC Service, having a third party service will make your mind at ease but of course, a reputable service just like Binance is using "Refinitiv" is decent one.
Why is a company using a third party service safer? I'd argue the exact opposite. Companies often tender this out to the lowest bidder, who invariably have very poor security. It also means your KYC documents are being shared around with more people and with third party companies, so there are more opportunities for a malicious employee to access them or a hacker to steal them. Your Binance example is a poor one considering they were hacked for thousands of users' documents and information just a few months ago.