maven wroteThe rule is the same at all levels: a free kick must be a place kick or drop kick, as BBR states. A special exception is made for free kicks after a safety, where a punt is an additional option.
The rationale for the exception is surely the fact that a free kick following a safety is made from the 20 rather than the usual free kick spot.
No, that's not the rationale. Originally in American football it was from behind the 25 and had to be a drop kick. Before that in rugby, it was from behind the goal line and had to be a punt. But later for a long time in the various American codes, it was not allowed to be a punt.
There really was never a rationale. These were completely arbitrary distinctions. There was a movement during approximately the 2nd half of the 20th Century to conform all the free kick rules to each other; previously the kickoff hadn't even been called a free kick. That movement receded and the rules partly diverged from each other again.