Update
Thanks to Sandra from Congressman Mica's office (Chairman of the House Aviation Subcommittee) there is some clarification on the TSA rule as it affects flight examiners.
TSA will email the training provider confirmation that a student has submitted all the required documents for background checks. THAT email must be shown to the flight examiner at the time of the checkride - not the results of the TSA's background check which are not disclosed unless there's a problem in which case a checkride is probably irrelevant.
The issue of starting training while TSA conduct their checks has yet to be solidly clarified. Will update after further talks with Rep . Mica's office.