"There is always the possibility of limiting immigration from outside the EU, which in my view may well have some traction, but May simply chose not to tackle it for the whole of her time in the Home office"
She didn't 'choose' not to tackle it, she wanted to (I have it on good authority) but it involves a great many difficulties, and I suspect she was under pressure from Cameron not to (though thats only my opinion).
And in fact the Conservatives DID reduce non EU immigration, just not very much. They also clamped down on bogus colleges acting as portals for illegals.
If you think about the categories that non EU immigrants fall under you can see why its hard:
Students
Family reunion
Highly skilled migrants
Inter company transfers
Asylum seekers
Which ones of those would you stop?
Personally I would love to see them preventing 'students' from doing any form of paid work (just as in the USA where they can only work on campus in a pre arranged job, not enter the general job market). Would you be happy with that? There would be an absolute outcry from FE colleges, Unis and English language schools however, all massive businesses