Primitive recursion seems to be related to bounded quantification. It is easier to make sense of bounded quantification with respect to natural numbers than to make sense of bounded quantification with respect to finite strings.
A relatively clean way out of this dilemma could be a two sorted language with both natural numbers and finite strings, and some basic functions between the two sorts, like the lengths of a string, or the string where one symbol is repeated n times.
A nicer solution could be to avoid the need for natural numbers, and work with some more native finite string related concepts, like head/tail, substring, ... instead.
I wonder whether there exist a robust computational model (on finite strings) corresponding to primitive recursion, just like Turing machines represents a robust computational model (on finite strings) corresponding to general ($\mu$-) recursion.