Ah bindweed.
There is some controversy about whether cardboard is too toxic to bring into the food garden. If you have a tall enough garden and thriving soil food web toxicity is not a big concern of mine. I would never put plastic in my garden. You are right that the garden needs to have contact with the earth to thrive.
I've been observing bindweed for about five years now and I've learned a lot. My preferred tactic is to wind up the stems and cover with a rock or brick. If you can go around every couple of days and tuck any new sprouts under the rock it will decrease the vigor of the
roots. In my
experience, trying to get all the roots out is only sort of effective, as the roots snap easily, the plant thrives in disturbed soil and every time you break the
root it secretes a hormone that makes it grow faster.
I've had success with beds deeply mulched with
straw on the beds and
wood chips over cardboard in the paths. You don't even really need a rock, you can just tie the bindweed up and tuck it under the mulch