Making a distinction between the actual mix and the data collated by the server hosting the website for a moment and focusing only on the mix element. If someone posted an address and then used what you refer to as a "mixer website" of their choice to mix, would you be able to find the exact location of the mixed funds?
Or, if someone already mixed using na website and gave you the post-mix address, would you be able to trace it back to the source? If the answer is yes, then that defeats the object of obfuscation the privacy seeker wanted and the/those mixer websites do not offer privacy. Did you ever test anything on the lines of this?
Your "if" statement isn't a possibility, funds are always associated with the original transaction when you use a "mixing website" since a "mixing website" has the exact same privacy as the traditional banking system.