The films and backing papers are designed together, because backing paper induced problems are a result of the emulsion and the ink and the paper interacting with each other.
If you rerolled the Kodak films with Ilford backing paper, you would encounter problems.
If you rerolled Ilford film with Kodak backing paper you might encounter problems too.
The economies mean that none of the film manufacturers can afford to have separate and different backing papers for each of their film offerings, so the Kodak stuff has to work with black and white negative film, colour negative film, and colour slide film, of all the various types.
Most likely it costs the film manufacturers more for to buy the backing paper for each roll of film than it does to make the roll itself. Ilford has admitted that publicly.