Re: Will it really?
Not quite. It’s airlines that, understandably, require documentation about who’s on their planes for internal flights, not the countries. I’m fairly certain I flew London - Edinburgh without going through passport control.
You can fly within the Schengen area without coming across passport control. Major airports like Amsterdam are even divided into a Schengen and a non-Schengen zone so you could do a connecting flight Toulouse-Amsterdam-Berlin without ever passing passport control.
As for the Irish Republic joining Schengen? Not a chance before Irish Unification. The other Schengen countries would never allow an open border with the UK and all 26 of them have a veto on new applications.