I understand your frustration with the subsidies that big corporations receive but I think your solution has some fundamental flaws because you're trying to fix it from the wrong end.
Let's hypothetically heavily reduce immigration, I don't think the corporate response to that will be to offer better t&c's to attract existing British people into the jobs. It will be to offer those same jobs on the same t&c's and where that doesn't attract a sufficient workforce then to reduce service levels and/or seek to increase technological support to reduce labour costs. You're right that too many big businesses and sectors are hooked on cheap labour but that's not a function of immigration it's a function of corporate greed.
In order to make those jobs respectably paid and to put into place the training programmes needed to transition the under employed into them requires significant investment. This will require a government compelling them to do so or changing the tax regime to facilitate the investment. So unless you're comfortable with the native population being exploited in the same way many immigrants are (which I don't think you are as your beef seems to be with the corporations?), then the only way to address this is by tackling the corporations themselves. There's no evidence to suggest that choking off labour supply and hoping the market adjusts itself in favour of the workforce will work. Evidence suggests the market will adapt in yet more dysfunctional ways.
My final point would be that economically immigration isn't a zero sum game anyway, the pie isn't fixed. However I think that's a bit academic to this discussion. If you want to fix the corporatism/extreme wealth problem you have to deal with the head of the snake not the tail.
You're just coming at it entirely from entirely the wrong angle, of course corporate greed lies at the heart of the issue, but it's mass immigration that has allowed that corporate greed and obscene wealth accumulation to increase vastly over the course of the last two decades.
Without the huge supply of cheap labour available through mass immigration the historically unprecedented stagnation and suppression of wages we've experienced in the last two decades simply couldn't have occurred, the same applies to the erosion of workers rights and T&C's.
It's really basic and immutable economics of supply and demand in terms of the value of labour, something that has existed since even long before Capitalism.
There's no evidence at all to suggest that removing that cheap labour the corporations will respond by effectively downsizing and reducing their output and the services they provide, that's a complete anathema to the way modern day corporatism operates and would leave them vulnerable to market competition for one thing.
In the face of cutting off their cheap labour supply the corporations could possibly attempt to limit their labour costs and maintain their obscene profit margins by going down the route of technological innovation... but why is that a bad thing?
It's very unlikely technological innovation could entirely counteract the exploitative use of millions of migrant labourers anyway so in some circumstances where technological innovation isn't an option the corporations would simply have no choice but to increase their pay levels.
There are other interlinked aspects to the whole situation of rampant corporatism and cheap labour, chiefly the utter insanity of a situation whereby we're importing millions of unskilled low paid workers whilst at the same time we have millions of working age people all under-employed through choice, CHOOSING to work minimal part time hours in order to receive optimum "in work" benefits even when permanent full-time hours are available to them... again all ultimately for the benefit of the corporations.
Your snake analogy works both ways, the way to deal with the head of that avaricious corporate snake is to remove its body.. in this case the snake's body is represented by the mass immigration that snake is exploiting and benefiting from so much.