I have lots of problems with this account. Yes, civilian ship radars work with transponders. They also register objects without transponders.
A MILITARY ship's radar necessarily works in an environment where hostile ships do not helpfully have transponders on. As a matter of fact, I would HOPE that the ship's sensor software would flag a ship with transponder off as "POTENTIAL HOSTILE"!
Was the captain "smoking a rope"?
Further, if her transponder was off, where did all these track recordings that we are seeing come from?