Oh dear… I was hoping someone else might chip in. Sounds a bad infestation.
Maybe a year of growing widely spaced plants which can be mulched again eg courgettes, squash, brassicas, even potatoes, combined with regular removel of bindweed stems poking through the planting holes. I hope you may have a cleaner area somewhere in the plot for growing salads and carrots.
Or grow on half the area and do a whole-year mulch on the other half, with some organic matter on top, to bring more roots nearer the surface, whence they can be pulled or prised out.
Wide bare paths around the edges are helpful as perennial weeds are always spreading inwards and the paths are a no mans’ land.
Good luck! and please inform us of how it goes.