IIRC what Kodak did was introduce a 120 format film with a smaller core to reduce the overall size, called 620.
Of course curling the film up tight played havoc with film flatness so the results were a bit dire and it never caught on outside the US (see also V8 cars the size of tanks). Kodak box cameras were so big any space saving was nugatory.
By the time the "disc" cameras hove into view the writing was thoroughly on the wall for Kodak anyway, whereas HP profited from its 96% gross margin ink for a long time.*
The "good" Kodak branded cameras, Retina and Retinette, used 135 like everybody else.
*The HP manager was a bit drunk when he came out with this, take it or leave it.