Let us just say for a moment that all of the mixing websites out there do not mix in the way that a coinjoin using Wasabi Wallet or Ginger Wallet does, in the end how is that detrimental to the aims of the privacy seeker if at the end of the mix the funds cannot be associated with the original transaction the privacy seeker wanted to obfuscate?
Your "if" statement isn't a possibility, funds are always associated with the original transaction when you use a "mixing website".