It's not fallacious and the UK did do that [0] for 7 years after they realised that the A8 migration estimates were incredibly wrong.
> In the UK, despite all of this immigration we have rather low rates of unemployment, marginally lower than before the 2004 expansion of the EU
First of all, no one here said immigration is causing unemployment, just wage stagnation for certain industries.
Second of all, it's not "counterintuitive", immigrants tend to move to areas with low unemployment so I would expect the above to be the case anyway. Most immigrants wouldn't come if there were no jobs.
> This soon hits the point in the modern globalised world that the industry just becomes uncompetetive and closes down, outsources, invests in labour reducing machinery, or moves
But that's not what's happening here, as you said, the jobs are still here (low unemployment). They're not automated or outsourced, they're done manually for minimum wage (or lower) by hard working immigrants from poorer countries.
I have to say, you kind of changed the topic. We were discussing about the existence of limitations for low skilled workforce, not whether or not Brexit was a good idea.