If it is just a conspiracy, it is not an effective one; neither is it just incompetence, if anything it is both.... an incompetent conspiracy!
Immigration for Britain was never a really 'toxic issue' until the late 50's when Enoch Powell; made it about about 'race'.
There have been plenty of examples over the years where properly planned 'immigration packages' have worked because the numbers coming were within a known/manageable figure, the reasons for their arrival were often quite clear they were either refugees escaping from war, or other forms of oppression, political and/or social and there was a process/procedure in place, that with proper forward planning could accommodate such immigration.
The main 'ignition-point' for the current malaise/toxicity over immigration was the 'freedom of movement' issues arising from Treaties imposed by the EU, that the British government had never sought approval from its populace, but where other EU countries did. This setup alternatives for many desperate people (and criminals) to exploit, once you got into the EU you could (in theory at least) move to any EU country you like.
Since then in the UK, with more and more people overstaying their permits, and an increase in turmoil around the world, successive governments who didn't want to (or could not) deal with these problems, lost their way completely, failing both would be immigrants, and those already here and opening the doors to a form of politics not seen in this country since the dark ages.
We are now about to reap the whirlwind, the 'Brexit debacle' was nothing compared to what is likely to arise, with Reform (moving us to the right) on one side and the Greens (the new left) moving us on the other flank.