Uk was one of the most powerful members of the EU before brexit and saying "one person (e.g. Angela Merkel) determine UKs immigration levels" is not correct.
UK is not part of the Schengen Agreement, like, Ireland they can decide if a particularity nationality need a visa, I have dual nationality and I can enter Ireland or Uk without a visa using one passport and need a visa using another.
Asylum papers aren't citizenship, and do not give the rights a EU citizen has (I believe typically they don't allow to leave the country), so the UK can freely choose if it wants to let them in or not.
> I believe typically they don't allow to leave the country
They do:
> You can travel to the so-called Schengen states (Belgium, Germany, Denmark, Estonia, Finland, France, Greece, Ireland, Iceland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands, Norway, Austria, Poland, Portugal, Romania , Sweden, Switzerland, Slovakia, Slovenia, Spain, Czech Republic, Hungary) without a visa, if you do not stay longer than three months and you do not work there. [0]
In the EU that's a moot point anyway as I haven't seen any border control in years. So at least travel-wise they're quite free to move.
UK and other Member States always had the right to decide for themselves with regards to immigration (movement between Member States was the only thing the EU influenced, and that was not up to Merkel or any head of state)
UK is not part of the Schengen Agreement, like, Ireland they can decide if a particularity nationality need a visa, I have dual nationality and I can enter Ireland or Uk without a visa using one passport and need a visa using another.