Shorting the windings doesn't really prevent the motor from moving, since it requires the motor to be moving to produce any resistance, so it really just limits the speed of rotation for a given force -- so it's a better solution for dynamic braking than static braking. A mechanical brake is generally a better solution for static braking, and it has the extra benefit of keeping the motor from moving even if the drive circuitry is disconnected. That, plus the fact that enhancement mode parts are far more common, with a wider variety of parts tailored to different power applications vs depletion mode at much lower prices (generally), is why you don't really see the solution you're proposing.
Using depletion mode parts in the bridge means you'll need a negative supply for the gate voltage, so the gate drive circuitry will be kind of unusual and would require a different sort of level shifting for the control signals. It may be difficult or impossible to use standard bridge driver ICs for this, due to the inverted drive voltage.
Single-channel gate drivers could probably be used with the right isolation/level shifting, so it's doable, but more complicated, and you lose the convenience and built-in protections of more integrated solutions.
As an alternative, you could use a standard enhancement-mode bridge, and then use a pair of depletion-mode part across the motor terminals. It's a few extra parts (including an isolated gate drive supply), but probably cheaper and easier overall.