Incorrect.
1. There is no evidence of the people the Romans called the "Scoti" (or the Gaels) before the 3rd Century AD, so there'd have been NO SCOTS to defend against in 122.
2. Archaeological evidence dated between the construction of Hadrian's Wall and Rome's abandonment of Britain shows the culture in the areas immediately north and south of the wall was identical. Which means culture flowed freely through the wall. And since culture only is transmitted by people, that further means the wall didn't much keep anyone in or out of Caledonia or Britain.
The better theory is that the wall was built to enforce Rome's tariffs on livestock and other trade goods passing between the two regions. The only places where it would have been convenient to cross with carts, wagons or livestock was at the Roman-controlled gates.
The wall worked the same way the other Roman border walls worked, controlling access and collecting taxes. On the barbarian side adjacent to the walls, Rome fostered allies, but the construction lines took advantage off geological rather than demographical boundaries.
https://freerepublic.com/focus/chat/4166604/posts?page=14#14