How can I use this weed?

I don't understand why people say this weed leaves biomass to the ground.I think the mulch of oxalis is very watery and gets evaporated leaving just thin strands,very very minimal organic matter.In the same  time it conquers most of other weed positions that leave something back to the garden and have great roots.

I wasn't aware this weed caused problems... It grows in my yard (6b in PA) but only in gaps between the rocks in my rock walls. The only natural predator I have noticed is myself and my kids. When they get big enough to bother with we rip them out and eat the top half.

It sounds like manual labor is doomed to fail. Move on to chemical warfare. No, not poisoning the stuff.... What you want is a tailored repellent. Many thingsgs (like poison ivy) won't grow in one or more specific soil profiles. Maybe change the pH a good bit one direction in a small area and change it the other way in another area. Add an irrigation system to a third area and so on. Change your soul to a fungal foundation elsewhere. Nothing grows well everywhere. Make a list of what you can change in your soil. Then change it. Take careful notes. If you find something that the weed hates, fine tune variations of that. When you get something the weeds can't be competitive in, find the vegetables that can be competitive, and run with it.