Hi Ladbrokes and welcome to the site.
I have found through the past year that it works best to have cardboard as TOP layer of a mulch. When buried under compost, it rots more quickly and weeds can then grow through it, whereas it stays dry and impermeable when on top, weighted down with a few stones, logs or whatever. Although after two or three months it may have weathered to the point of needing another layer of cardboard on top.
Black plastic is easier in this respect and can sometimes be scrounged from someone who has finished with a sheet.
In your case I would dig out the few remaining brambles, as it sounds like you have sorted most of them, and I would keep pulling bindweed to stop it regaining strength. I know of no ‘cure’ for bindweed, except to keep on top of it. A thick mulch certainly helps, but won’t kill it. Shoots that do appear are easier to pull out or remove with a trowel.
Weeding is a job which never goes away but after taking time to improve and clean soil, it becomes much easier, and I hope you keep moving in that direction. The first year or two are always the hardest when taking over infested soil.
Can anyone else offer some tips that have worked?