Thanks for your question re the removal of this native plant, Rubus spectabilis, commonly know as salmonberry. Although this plant plays a major role in forest habitat, it may not be wanted in a homeowner landscape. The following page from USDA/US Forest Service gives you some basic information on its removal and some more background on what you are dealing with:
Control of Salmonberry: Because it spreads rapidly and forms dense stands after overstory removal, salmonberry is considered a weed in some management contexts (i.e., after logging) , and much has been written on methods for controlling it (e.g., [94,230]). Because of its extensive rhizome network, salmonberry is difficult to remove once established. When top-killed, plants sprout from the stumps, root crowns, rhizomes, and/or aerial stems [294,296]. Salmonberry sprouting ability may decline after repeated, successive (i.e., monthly) top-kill , or when top-kill occurs between May and June . Simulated response of canopy cover and height to manual cutting was compared at each phenological stage: establishment (early spring bud break in May), reproductive (early summer fruit set in late June and early July), and senescence (fall, October). The longest-duration reduction of cover occurred following cutting at the reproductive stage .
From the OSU publication, "Salmonberry", https://www.google.com/url?client=internal-uds-cse&cx=001157565620839607635:f_5ovr-jasm&q=ht... you will find this statement discussing control by pulling and using herbicides:
If already present, salmonberry sprouts can be sprayed, manually cut, or pulled out of the ground, although cutting and pulling may need to be repeated. In pure stands of salmonberry, the herbicide glyphosate can be used from late summer to early fall to damage or kill the plants.