The whole admitting fault has nothing to do with the ATC side of the investigation. There’s no pilot statement on their end at all. Their data collecting process during the Brasher phone call is nothing more than pilots name, address, cert number and contact information. Done. They don’t do compliance actions nor enforcement actions.
Even if you just happen to tell them over the phone and admit you were in fault, theres no requirement to log that. Now, if you admit fault to ATC and it gets logged in the remarks sections, the investigation process on their end doesn’t stop. You could’ve actually been correct and the controller wrong phraseology. That’s where the facility QA comes into play. ATC QA will still collect data, review data, then decide if it’s an actual PD and warrants being sent to the ASI.
Now, the ASI will ask questions and basically admitting fault will enter into the equation. Personally, since the vast majority of inadvertent PDs will result in a warning letter or remedial training, I wouldn’t be worried about admitting fault. Even if it results in a pilot suspension, unless you’re looking to fly professionally, who cares?