“…so the cost of patrolling borders will sink to zero”
You’ve not quite got the gist, Peter.
The cost of patrolling the UK’s borders is not the issue. We’re being asked for additional funds to patrol the external borders of the Schengen Area, of which we are not part. The EU, in its utter stupidity, effectively extended the borders of the western European countries to Russia, Ukraine and Turkey. The western European nations suffering the results were complicit in that folly by signing up to Schengen. The countries on the eastern end of the bloc (principally the Baltic States and Greece) are singularly unable to protect their borders. But instead of incursions to those countries being confined to them by robust internal borders, any illegals making it to the eastern EU nations are immediately able to roam wherever they like.
The Schengen Agreement has exacerbated beyond belief what is already a huge problem. Its designers were adequately warned of the perils but those warnings were brushed aside. The UK (wisely for once) opted out of Schengen but is now being asked to contribute to solving the second grandest folly (second only to the euro) of modern political times. So long as we remain members there is no way the UK will avoid such payments. In fact, it won’t be too long before the Schengen opt out secured by the UK and Ireland comes under threat (as eventually do all such opt-outs; it’s the way the EU works: "we'll let you off - for now"). Leaving is the only answer.