It was President Juncker who said he thought borders were the worst invention of politicians ever, not the Brits who we learned recently are rather fond of borders. Your argument is backwards.
Rejecting all borders is a tenable ethical claim - indeed it's hard to justify why states should exist in most ethical frameworks that value human equality - but Juncker didn't go there.
But yes, Britain sabotaged the EU, including its having a joined-up defense and border strategy. Indeed, you might expect the departure of the most Eurosceptic nation to strengthen the EU.
But hey, if you want to believe these people think differently to how they actually do, go ahead and believe that.