Did we have too many "wee drams" today?
And that gets me back to why I said "don't confuse Schengen with the EU."
Like you said, the UK and Ireland are part of the EU but not part of Schengen. Switzerland is not part of the EU but is part of Schengen.
So, if the average commoner thinks of them as two separate things, it makes life much easier. And don't we want life to be easy?