I believe the minimum depth without conduit would be 3'. Instead of USE, have you priced UF cable? (I haven't. Just asking.)

Two feet actually.

Can't comment on the best wire without knowing how much load you need at the barn and how far the feeder run is going to be.

I only want a clean 40A at the barn. I've done the voltage drop calcs including 225' run of SE cable running though another barn. The underground run is 225 ft. I'll need 1/0 for the hots which will give me 3%  total drop at the pole barn. I was planning on something smaller for the neutral, maybe #2, as my plan is to use 20A of the load for a 240V lift. I'll use #4 for ground. I can't find the assembled cable similar to this. I read NEC at my leisure today and it seems to allow approved wires (not cable) direct buried. I have a backhoe and plan to bury plenty deep.

I'm leary of using "something smaller" for the neutral. Both the hots and neutrals have to have the ampacity protected by the breaker. The neutral plays its part in voltage drop for 120V loads as well.

Unless I'm misunderstanding something here, You should be able to get away with the readily available 2-2-2-4 USE.