Canadian thistle has rhizomes. Even a tiny bit of that root left behind will regrow, forever and ever, and they're much too deep to effectively dig by hand. Conversely bullthistle is a biennial with a taproot, and all you need to do is sever the root a few inches below the surface.
Repeatedly plowing and turning the soil while it's very hot might dry out enough roots to be helpful, but you'll lose as much topsoil to wind as you'll kill thistle, so not the best course.
Roundup won't kill them entirely but it would give you a head start. Same with hand-digging what you can reach (and remember to pull as much root as you can get out). However R Scott's remark about pH gives me an idea... what if you saturated that patch of ground with sulfur, enough to turn it really acidic? would the thistle survive this abuse? might be worth testing the soil and trying sulfur on one little spot. As to what good shelterbelt trees would like that ground afterward, maybe spruce, and do remember it will run off somewhat into adjacent soil.
You'll definitely want a tree that hugs the ground at the bottom, tho (which spruce will do if not planted too close together) to starve any remaining thistle of light.
A number of plants produce toxins of their own to discourage competition, and some of these (including that from black walnut) are a whole lot more toxic than Roundup, and far more persistent in the soil (Roundup was developed specifically because a product was needed that broke down very fast and did not remain significantly toxic). I wouldn't worry too much about doing a spot application as needed.
Remember to plant your
shelter belt as several staggered rows of trees, not too close together, or eventually they will thin out at the bottom and that kinda defeats the purpose, even if they don't kill each other competing for water. My aunt told me that they had to keep the grass plowed down in their shelter belt or it would kill the trees. She was up on the Teton bench, which is a bit south of and basically the same climate as yours. She had mostly Russian olive and carragana, which are both pretty tough.