Integer promotion will not help, because it may not go to a sufficiently wide unsigned type to cover the shift. (In practice it will, but unsigned int could be just 16 bits).

Promotion of unsigned chars to unsigned int would have problems of its own, mostly because unsigned arithmetic (modulo power of two arithmetic) is inappropriate for most uses, and error-prone: it has a large, silent discontinuity right next to zero.

Alas, in fact, unsigned chars can promote to unsigned int: on rare platforms like DSP's where you have sizeof (int) == 1. Sigh.