You are describing exactly the issue I am trying to shake out (no pun intended) in my 2013.
Mine was in an accident, and the steering shake was the first thing I noticed after getting it back from the body shop, since the rental did not exhibit this trait. The body shop and service department both test drove it and said the shake was normal. They fully inspected the front end and did not find anything wrong with my steering rack or front suspension.
I have since found the below posts describing this issue on other WK2s, one of them a 2014. At this point my opinion is that the steering intermediate shaft and/or shaft bearing are at fault for this negative feedback characteristic.
2014 V6 4X4 Steering issue
Feel road in steering