Let me start by saying I love my new Tucson SEL Convenience. Only concern is a little bit of bucking when it first starts out and I seem to occasionally smell something like transmission fluid or bearing grease. Let's not get into my ongoing issues with the Android smartphone digital key app.
With reference to your vibration, I had a new 1988 Audi 80 years back with the same problem. Went through multiple balancing's, alignments, new tires, front end checks and still the problem persisted. The Audi rep came and sat next to me while the steering wheel was shaking profusely and claimed it was "normal road handling"! They dragged things out until I could no longer qualify for the Lemon Law (watch for this). I concluded the car must have fallen off the truck at some point resulting in mechanical damage to the frame or steering. I eventually walked away from the car, it was a lease, and suffered the consequences. I hope they can resolve this issue for you because the car really is awesome! Good luck!
Wow! Unusual for Audi. When the Audi rep said "normal road handling", I would have obtained a lawyer the next day, knowing this is NOT normal anything!
Yeah, we love the Tucson also. I will assume ya read all the posts here. Seems to be a common problem, this steering issue. The tech at my dealership let it slip that many others were having the same issue. "We" are upset at the lack of expediency to resolve the problem, and the lack of usual diagnostics for this issue.
As for me, Hyundai is running out of time, quickly.

Just a quick note....my son has a 2015 Audi S6....we tuned it and put a big turbos (2) kit in it.....800hp....0-60 in 2.4 seconds, 70k miles on it. So far one stupid exhaust sensor has failed, not bad for beating it to death. Sorry you had bad luck with them. Audi USA has been very helpful to us.