I can toss my $0.02, my family came to Illinois from Wurtenburg circa 1850 (both sides) and settled in the central part of the state. My grandmother was born 1888 and learned to make kraut from her mother. Kraut and cabbage as a whole was big in the family food prep. Grandma said you could tell where a person came from by how they fixed their cabbage rolls. Wurtenburg being south east, grandma used tomatoes and caraway seeds in her cabbage rolls, further north you went, people used kraut to cook the rolls.
I am pretty sure the Hessian soldiers employed by the Brits had kraut, so I am guessing it was present from the 18th century.