It appears that you (and most of the folks above) have the same confusion that I had when I got my Nautilus, which has the same systems as in your Aviator. There are TWO systems here that do different things yet work together:
1. The Lane KEEPING system. This system is switched on by the button on the end of the turn signal stalk and will provide vibrating feedback and aid if you drive outside the lane markers in the event of an unintended lane departure. In your second video, you set this feedback to Alert + Aid and sensitivity to High. When it is active, you will see green rails on the display.
2. The Lane CENTERING system. This system is switched on by the button on the right side of the steering wheel with the little steering wheel on it. You mistakenly called it “Steering Assist“ at the 1 minute mark in your first video, understandable since its icon when active is a small steering wheel. This system is designed to keep you driving in the center of the lane, and will make small corrections in the steering to keep you there once the system is engaged (Shown by the small steering wheel icon in the instrument panel display). Fun fact: the reason the Lane Centering System has a button on the steering wheel and the Lane Keeping System does not is because one requirement for activation of the Lane Centering System is having both hands on the steering wheel.
Now that I hope you have a better understanding of these two systems, I suggest you go back and review the sections in the Owners Manual that describe each of the the Lane KEEPING system and the Lane CENTERING system. Pay attention to the specific requirements for each system to be functional. I say that because your videos were taken with the car in your garage, but the systems need to see road markings to activate. That‘s why things were not responding the way you might have expected.
You might want to bookmark the online version of your manual, which is searchable.
Search for KEEPING or CENTERING and you will find all the relevant information for each system.
Hope this helps. P.s. I like your cat.