"You can easily tell people's immigration status"

My point was that you have no idea what status a person holds. They could be citizens who prefer to speak a different language. They could be on holiday. They could be students. They could be in the country without permission. They could be anywhere on the spectrum of immigration permission. You don't know their status.

Besides, I went to a shop the other day and not a single person was speaking Irish. They were CLEARLY immigrants.

Let me turn that around: if someone is an immigrant forever even once they have achieved indefinite leave to remain, in what sense is it even possible for them to assimilate?

How are you turning that around? You're just misunderstanding the meaning of the words assimilate and immigrant.

You're an immigrant till the day you die if you move to a foreign country, but you can integrate with your host country.

Immigrant definition: a person who comes to live permanently in a foreign country.

Someone who lives temporarily is a migrant.