Does the code permit the ampacity of switch legs to be less than the ampere rating of the branch circuit?

No, and here's why: the rating of the branch, determined by the overcurrent protection, is designed for the ampacity of the branch conductors so that they breaker trips before the conductors heat up enough to start a fire. If you put in smaller wires on the same branch, they will get MUCH hotter before the circuit breaker trips, if it ever does.

For example, a 20A branch, protecting 12AWG wiring; add some 14 AWG rated for 15A. Something faults the switched leg at 18A; enough to burn out the wiring but not enough to trip the breaker. House burns down, insurance doesn't pay because of "faulty wiring" installed negligently by a person without proper training and credentials.