Thanks for the advice, everyone! Sounds like a combination of redistributing the deepest areas and adding nitrogen, with composting and/or cover cropping, is going to be the long-term fix. I'm glad to know that soil contact will help it break down quicker; some of the plastic was tricky to get out because of tree and rhododendron roots growing over it, so those are definitely areas where I'll be composting it in place (and hopefully improving the health of those big plantings).
Roberto - Is there a screen size you'd recommend? I've got some 1/2" square wire mesh left from making compost hoops; would that work or is it not the right size? I think that even with redistributing the bark, I'm going to have some left over, so I might as well get the biggest pieces out into their own pile.
Marco - Where I've amended the soil, the rain actually sinks in. Everywhere else, whether in the beauty bark areas or where bare soil is exposed, the water gets in maybe an inch, then sheets off. Where the pure native soil is, the only way I've gotten water into it at all is to water slowly while "stirring" it with a cultivator fork. Otherwise, the water literally beads up and rolls off with a little puff of dust behind it. Even rainwater takes a long time to soak in. Guess I've got a lot of raking and hoeing to do...
Thanks again! I'll definitely be raiding the nearby coffee places for used grounds. Not ready for chickens of my own yet, but my neighbors have some, so perhaps they can spare me some used bedding.