We bought a house this past spring and when we toured the house in March the backyard didn't look like much more than dirt and weeds. When we got the keys in May, the entire backyard was full of ground cover. I finally identified it as houttuynia and after reading about it online, I now get depressed every time I look in my backyard. I want to to remove all of the houttuynia and have grass. I know this won't be easy, but is it possible?

I'm looking to dig up a 25ft x 40ft space. The outer border of the yard has mulch and some bushes which is fine. From what I read and given the extent of the plant growth, I don't think round up is an option. My plan is to dig down 1-2 ft, remove roots & soil, and then fill with new soil. I'm contemplating using a landscape fabric to prevent new growth, but most people don't seem to have positive experiences with fabric. Any other suggestions?