I'm going to have to look up what bindweed is. I wouldn't know what to look for. Yikes! I hope you can get it under control, as it sounds nasty.
Okay, I looked it up, and I can see why you can't get rid of it. You have to broadfork it deep to get all the roots. It did say you could apply glyphosate to the bindweed in the fall when the bindweed is actively growing. And, use Preen in early spring to keep the seeds at bay. Sounds like a lot of digging for you.
I refuse to use glyphosate in my garden, so that's not an option. But I just found another article online that suggests cutting it off (repeatedly) at the ground. That's supposed to eventually discourage it. I think I will try that next.
I don't blame you for the glyphosate; I wouldn't want to use it either. I hope it works for you, giving it frequent haircuts.