"you'll no longer be covered by the European Health Card co-operative agreement,"

That rather depends upon what is agreed for brexit. Like many issues it is a two way street.
If a uk driving permit is no longer recognised in Europe all those foreign cars and trucks will evaporate off british roads and Ireland's tourism will be massacred,or come at a much higher ferry price
(I suspect300% increase in ferry cost eg Rotterdam dublin 650 miles, plymouth santander 447 miles)

Likewise demands on the uk health services by eu nationals will require proof of funds prior to treatment unless sense prevails.

You seem to forget many organisations predate the EU. Nato and interpol being but 2 examples.

Do the EU really want to cut off their nose to spite their face?

Stupidity will inflict pain on both sides, something the negotiating wannabe bullies of the EU negotiating team seem to forget.