Why does Brexit have to equal "increased bureaucracy at the border"? Is it due to the lifelong bureaucrats that loves enforcing it? An effective government should do everything in its power to reduce the bureaucracy in extraordinary times.
Except countries in the EU, which has a common market that made such checks unnecessary inside of it. Of course there are still customs on the outside EU border.
So if you leave that, and your borders are now not internal market borders anymore but your and the EU's external borders, you're going to do more customs checks than you used to have. In both directions.
I agree with you that bureaucracy here is the problem, and Brexit was the cause of it.