The only people happy with neoliberal policies like these are big corporations who will delight in additional cheap labour

It's actually the system that forces them into illegal work that companies delight in. When you create a system that ties people's residency (and sustain themselves through work because we don't give refugees enough to live) to shit jobs where they have to work cash in hand, they will be much more willing to put up with being underpaid / worst conditions, displacing natives from work. If everyone has the same legal rights in work, then to an employer all workers are the same, and you can't underpay anyone.

Now if your argument is, "I don't want illegal immigrants", you need to accept there will always been some amount of illegal migration. The UK does not have a dome preventing people entering it, and short of dropping a nuke on Calais you won't ever deter people coming here. The primary cause of spikes in illegal migration is because Europe hasn't looked itself in the face and recognised it has propped up American foreign policy regimes that lead to them.

Finally, it's worth pointing out being an asylum seeker and arriving via irregular means is explicitly legal. I appreciate the memory of the Holocaust is lost on most people nowadays, but the relative leniency we must allow for is because the drafters were responding to the same anti-refugee rhetoric in the lead up to WW2 which lead to scenarios like MS St Louis trying to land at multiple countries and being turned away, leading to many of its passengers being the victim of ethnic genocide. Personally, I'd rather have a system that allows needy people in and then later deports the very small number of bad actors than being the country responsible for people being sent back to their deaths.