If it's just a thin layer direct onto soil then I'd agree, just dig it in.
If there's a membrane, you can get hold of a corner of the membrane and slowly pull it up, dragging the chippings as you go. You'll reach a point where it's too heavy to lift any more by when they'll be deep enough to shovel into something - bucket, wheelbarrow - and move them to a heap in a corner. Then cut off the membrane that's now clear and start planting whatever you had in mind in the clear section. Then pick up another bit of membrane and do the next section. If you do it over a long period of time, planting behind as you go, it'll help you to focus your planting on a smallish section at a time - which is no bad thing - knowing the bits you've not got round to yet won't be getting weedy and overgrown
Use the cleared chippings either for dressing pots, or mulching, or where you want paths, leave the chippings there, possibly adding more if they aren't very deep. Or sell them on gumtree a few bags at a time.
You could form beds or lawns using the soil in the raised bed the same way, a few wheelbarrows at a time.
You will make it all much harder to deal with if you try to do it all at once.
Think of your new garden-to-be (how exciting) as a process not a thing you have to do one weekend. If you have an overall plan, you can gradually bring it into being; you don't have to blitz it.