We've done this.
One year I covered the kill area with black plastic and let it sit in the sun for a a couple of weeks. I did that once it got good and warm. Worked great. If you're trying to prep new beds now, it might not get hot enough under there to kill all the annual seeds.
One year I covered the area with black geo cloth and cut holes in it for the new plants. That worked, too. Then we mulched and I added new plants, but had to cut into the geotex which was harder than just digging a hole. We used this for my raised beds and it worked great. FIrst I skimmed off the layer of grass, then covered with geotex and put the bed on top. 12 inch deep beds. Full of sand, leaves, chicken dirt, etc. Best veg ever. Didn't have to mess with the geotex under it because it was so deep.
We've used round-up but it will take more than one spray. First the perennials come up and you spray them, but then the annuals come up and you need to re-spray.
I'm sympathetic. We're on clay, too.