Woody stuff needs lots of nitrogen to break down so is best composted elsewhere then used on the garden after a couple of years when it will have broken down a bit more.
The big problem is perennial weeds - annuals are fairly easy to deal with.
The method we have used here with excellent results, is to lay a very thick layer of old straw or bulky fresh FYM over the bed, then cover that with a light excluding material such as old woollen carpets (although who has them to chuck out?) black polythene and so on. Black polythene does let some light through. A tarp over a thick layer of straw doesn't - the deeper the better. You can put opened out cardboard boxes over the straw and under the polythene, but be sure to removes any sticky tape which doesn't break down. we left ours for a good year, after which it was beautifully soft and crumbly and ready to plant potatoes into.
Perennialweeds do come up under the light excluding cover, but they are thin and white and eventually die off. If you feel there are still perennial roots in when you come to plant, dig them by hand at that point - the soil is so easy to work. This year we have planted our potatoes through a black light excluding fabric - this would be particularly useful if your initial go hasn't totally got rid of the weeds - so the ground will have a second chanced to get rid of those leaves but you still get a crop from the ground.
Similarly, annual weeds do often germinate under the light excluding fabric, but don't survive a season in darkness, so you start with a lovely clean spot.
So yes - it's definitely worth trying the thick mulch layer approach.
If you cover your plot, it won't be completely cleared by the start of the next growing season.