Painting herbicide on the stumps may work. There is an interesting applicator that looks like a golf putter you might find handy. The toxicity issue is controversial--lot's of strong opinions. Round-up is considered safer because it doesn't persist in the soil. But I'm not sure whether it really works with bramble.

I don't use herbicides, and the reasons are not really a "philosophical." It's mainly because I'm not so confident I'll follow the instructions to the letter, and that's a must. Be sure to read and understand the danger of contaminating water sources. Round up is quite toxic to marine life. How you use herbicides is the number one key to safety.

The best bramble control I've found is 2 or three mowings during the growing season. In my experience I've sort of hacked it back to an acceptable extent. But it still leaves ample areas of bramble growth, and I've come to love them.

Because bramble is a "frontier" growth, there's actually more species diversity than first meets the eye. If you can imagine some of the areas as being eventually wooded--here in the East that's what the land wants to be--you can help that process along a bit if you want. In some areas of bramble that I haven't mown, Black Walnut trees now stand about ten feet or more. It won't be too long before these areas are more wooded than bramble.

You can kill pretty much any plant with darkness, using dark black plastic combined with patience. I've done blackberries, ivy and dandelions with black plastic.

Blackberries are a challenge for this message, because you have to find and kill all the spreading bits, but eventually, it'll work. Check every week or two under the plastic with a flashlight so you don't encourage growth when you peek.

Clear plastic can also work, but for different reasons - it will fry plants like bugs. But in the rainy/dark part of the year, I think black plastic is more reliable.