The worst I had to deal with was pampas grass, which appears to be a plant made of actual swords.

I spent three days hacking at it in a coat so I wouldn't get shredded. When I finally cut the root bulb out it was a cube of wood a foot across. I could barely lift it out, I had to roll it to the bin.

At least pampas grass doesn't spread.

I don’t know if this will work on brambles, but for pesky root systems I’ve had luck with Bonide’s Stump and Vine killer. You cut near the base of one of them, then paint the exposed stem with this stuff. It absorbs into the root system and kills all of it. Works great on pokeweed.

Edit: Turns out this is just a specific brand of triclopyr herbicide like MoonMelon mentioned. So here’s another recommendation for triclopyr!

I don't know if you're opposed to herbicide, but triclopyr will kill it. You can get triclopyr salt (water based) and apply it to the freshly cut stump surface (within moments of cutting), or triclopyr ester (oil based) and apply it to the outside of the plant close to the base, no cutting required. Both of these will kill the root. Otherwise just keep cutting and eventually you will exhaust the root.

Not sure about elsewhere, but in the USA you can typically buy the water-based triclopyr salt in a small bottle with a brush attached to the cap. This is in pretty much any garden store. Even though you have to cut the plant first I think this is the best form for just a few plants.

Brambles can be valuable plants, providing shelter and food for many small animals and tasty blackberries for people. But, if they become noxious, they can spread quickly and choke out all other plants. They spread by rooting from the plant tips and even if you dig up the root system, any little piece of root can and will re-root and grow a new plant.

Either move the shed to get at it - all of it - or you honestly may need to resort to herbicide to kill it. It sounds like you have fought them mechanically and are losing the war. I would recommend consulting your local garden center for the best herbicide to apply to kill them.