No suggestions, but I also have some to get rid of so definitely following.

Purple Bindweed is a Texas native...that grows aggressively in the heat, but dies back fast when it dips down to the mid-20s...

Oh dear. I recognize that by its bloom. I had no idea THAT was Bindweed. I appreciate your posting the link. You just have no idea how disappointed I am now to know that it's everywhere.

^ Well, there's a few different types. But that purple one is native and I'm actually testing it out this year. I'll see how it does over a whole season...but it seems like its fast growth in the Summer gets counteracted every year by its fast dieback in the Winter. So, I don't think it will just keep progressively taking over like Trumpet Creeper or Wisteria, etc... I believe it's more like an annual than a perennial.

Purple bindweed is a perennial morning glory (Ipomoea cordatotriloba) that can become well established in fairly low light, so roots are often too extensive to pull completely by the time they're noticed growing up through large shrubs or perennials. Small root remnants will return and spread in any site where they're happy. It also volunteers fairly reliably when left to seed.

In our yard, it does seem a bit more particular about location than the trumpet creeper (Campsis radicans). However, well established vines of either are robust enough to take multiple seasons to eradicate where happy. When digging up the roots won't or can't work, I resort to herbicide delivered by floral tube. It's an effective way to provide a well contained concentrated dose to only the target vine. Of course, not an option where kids or pets might be tempted to mess with them; our dogs leave them alone, but would not assume all would.

This tube is out for the photo after refilling - normally it would be much less obvious left standing in a small hole dug into the bed. A glove (or two) helps avoid skin contact while handling and tubes should not be reused for anything other than herbicide applications.