Schengen is about being able to check at your borders. Indeed the UK can still do that, despite being in Europe.

However, the UK still has to allow free movement of people. Meaning that it can not stop Europeans from living in the UK. This does not cover working though, and I believe that some European nationalities still have to apply for a work permit.

See http://www.europarl.europa.eu/factsheets/en/sheet/147/free-m...

UK can stop anyone who is deemed national security from entering(if the worry is criminals coming here) and free movement applies only if you're employed, a student or can demonstrate that you have enough funds to live here without assistance - without it you can be denied entry. So that should solve the issue of "immigrants coming here for benefits". Really, UK has full power over its borders, it just chose not to exercise if.

You get it backwards, free movement of people is for workers, you don't have the right to stay if you aren't working (or alternatively studying, or living with someone who works) — though this isn't heavily enforced

You're right. I just looked it up to be sure. When I lived in the UK in 2007, Bulgaria and Romania entered the EU. There was some fuss about that, and initially those nationalities did not automatically obtain a work permit.

These were granted the same rights in 2014.

https://www.bbc.com/news/world-europe-25237742

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.

See my reply above - even with free movement, there's plenty of cases where you can deny someone entry. Right to free movement is not absolute, and UK is in a fantastic position because it can inspect every single person coming here, unlike for example Germany.

To clarify the united states system, states have rights to regulate commerce within their territories. So movement of goods is not in fact free. For example, California has border gates where they will sporadically check for and confiscate certain agricultural products.

Yes, the state that wants to be a sanctuary state has border gates and enforcement.

Indeed, and it doesn't. Non-EU immigrants still outnumber EU immigrants.