EU has precisely zero stakes in Brexit.
That's wrong in one major and important way: the EU had a vested interest in making Brexit as difficult as possible, to dissuade any other member states from thinking that leaving the union would be easy or desirable.
That's why the EU was so difficult in a lot of the negotiations and wanted to make the issue over a possible border with Northern Ireland so hard, while at the same time making sly comments about how they would "love to have the UK stay" etc (to try to sway the undecideds and win over the Remainers - although the latter wasn't hard, since in the eyes of many of them, the EU can do no wrong).
However, that's also why IMO the UK Brexit negotiating Tories were so unwise to talk and act like they were playing "hardball". They should've been smarter and more careful than that, knowing that the EU negotiators (people like Michel Barnier) were trying to make the departure of the UK as difficult and awkward as possible, whilst on the surface trying to sound the more reasonable and understanding parties.
The best way to counter someone who is trying to manipulate you is to smile sweetly, be as nice as possible in return, but then play them at their own game - not try to act and talk like the toughest guy in the room and butt heads in a misguided attempt to make the other side back down. In that sense, the EU negotiators played a much more crafty and well-planned game (apart from Juncker who occasionally got caught voicing his real views). But then, the UK also had the major (self-inflicted) disadvantage of having first Theresa May (too weak) then Boris Johnson (well, we all know how well that turned out) as leaders at the time!