Can a controller clear you through an SFRA?
An AOPA article on this has a chart of the Leesburg Maneuvering Area (LMA) which, as you said, doesn't require a DC SFRA flight plan (you do need one for pattern work, though):



If you compare the area in the chart to the flight path in the image in your question, it looks to me like Joe entered the SFRA southwest of JYO and then entered the south edge of the LMA. I suspect the FAA is saying that he should have flown further north, then entered the LMA from the west. In other words, he cut through a corner of the SFRA. I've marked that in red here:



The LMA NOTAM says:


  PILOTS MUST ENTER THE LMA VIA THE MOST DIRECT ROUTE AND AVOID ENTERING
  REST OF THE DC SFRA


By cutting the corner, Joe entered a part of the SFRA that was outside the LMA, hence the call with the FAA. That's my interpretation, at least, although I'm surprised that the FAA didn't spell out to him very clearly what the issue was.

And to answer the question in the title, ATC cannot clear a pilot to do something he isn't authorized to do. ATC could clear a student pilot into class B airspace, but if the student doesn't have a logbook endorsement for that airspace then he still isn't allowed to enter (barring emergencies). "Proceed direct Leesburg" doesn't mean "fly in a straight line to Leesburg regardless of other considerations", the pilot still has to get there legally.