if there are stumps outside the deck, and a massive cutting done, I'd say you will have a massive problem next year if not sooner.

You will likely have to consider it a long-term project. If you really like the home, do some more research and consider the landscape issues in your offer. Check wikipedia and other kudzu websites for helpful info. too.
Unfortunately deer don't eat it, nor many other invasives for that matter. In past years, to restore a small forest, I have cut the vines at the ground, and pulled whatever roots I can. If I cannot pull the root out, I paint the cut with herbicide. I've read that this is especially effective in the fall, when the plant is sending nutrients back to the roots for winter.
Since I have been eliminating the kudzu, however, invasive grape vines took over instead. The grape vines have crawled over the tops of all the smaller trees, eventually starving them of light. Then the weight of those vines bring the dead trees down and the vine actually re-roots there. When I finally found the main root, it was as thick as my thigh; required a chainsaw.
It will be easier to pull roots when the soil is wet, after a good rain. Also, every Fall, I added several feet of mulch (leaves donated by neighbors with large oak trees). As they decomposed, this helped to loosen (and enrich) the soil so that when I did pull roots, it was much easier.
Once you eliminate the invasives, you will have to keep after them year after year, to eliminate new sprouts from roots and seeds. Over time, and with patience, you can encourage native plants and trees. But it takes time and patience and work. Owning a home and keeping the landscape nice is a lot of work, but it can be very rewarding.