Earlier quoted context omitted.
Which was dumb then and still is - UK was never part of the Schengen agreement and has full control of its borders, it can stop anyone from entering as it pleases.
Not really, no. One of the basic tenets of the EU single market is free movement of goods and people. Schengen agreement just removes border controls. UK has the freedom to stop any non-EU citizen from entering as it pleases but so does every other EU country, including the Schengen ones.
Yes, the state that wants to be a sanctuary state has border gates and enforcement.