"Britain, having voted for Brexit partly as a means to control its own borders, can't really complain if the EU wants to do the same."
You missed a bit out and misunderstand another bit:
(1) The United Kingdom voted to control its own borders as it thinks fit. Not as unelected foreign bureaucrats think it should.
(2) The EU does not have any borders to control. They are under the control (for the moment, at least) of the individual nations.
The Irish border is a non-issue. Billions of pounds worth of goods currently enter the EU (from outside) via such places as Dover, Felixtowe, Tilbury, Gatwick and Heathrow every year. Less than 1% of it is physically checked. Similar arrangements prevail in other EU countries, including those with land borders with "the great beyond". But suddenly the piffling amount of trade (relatively speaking) that crosses the Irish border is seen as an insurmountable problem.
But let's assume for the sake of argument that it is truly insurmountable. The UK has no intention of imposing a hard border and neither does Ireland. That is a fact. The EU has neither the resources nor the authority to do so. So, I'll ask again: just what will happen on March 31st next year if the UK leaves with no "deal" agreed? What will happen to lorries travelling from Belfast to Dublin?