Shakey steering wheel.all speeds one brand new tire.3 pretty worn.front passenger has small slits in the inside sidewall
Its almost as if you have answered your own question here. The most common reason for your issue would be related to your tires. A test you can perform would be to swap the front tires with the rear tires and see if the shaky steering wheel changes and turns into a shaky feel in the back of the vehicle.
Busted Belt in tire
If your tires are worn it would be in your best interest to replace them. Keep in mind that the problem tire could still be the brand new one, not as likely but still possible. What happens is the internal belt on the tire starts to separate. This causes the tread to become uneven. And the uneven tread will cause a vibration at all speeds. This is referred to as a busted belt in the tire. A little as hitting a metal plate in the road in a construction zone or a pothole can cause this to happen. And can happen to new or used tires.
Reddit has a nice video of what I am picturing when talking about with the steering wheel shakey at all speeds.