Sterling is an EU player because of his Jamaican passport.
The Kolpak decision (linked in my previous comment) means countries who had specific treaties with the EU nations count as EU players and that includes most of the Caribbean including Jamaica.
I think you have massively misread that. Happy to be corrected. It extends EU rights to people living and working in the EU if feom countries that have EU association agreements. Jamaica may ot may not, but he isn't a jamaican living in the EU, he is in the UK.