Oh, they will do exactly that if it means preventing losing the whole head.
This is not a matter of pride here, it is a matter of self-preservation.
People argued that the EU won´t take a hard stance in the upcoming trade negotiations because the countries won´t do something against their own economica interests, but at least when it comes to the major players of the EU their biggest interest, both in the political and economical sense, is to keep the remaining EU intact. This means keeping their own EU sceptic parties at bay and discourage any EU member following the same path the UK just chose.
If they let off UK easily in the trade talks, this will start a chain reaction of other countries leaving the Union which might put the whole thing at risk. And that is at least from a German perspective a very dangerous thought, as we are both the biggest contributor and profiteur of the whole thing.
The prime interest of the EU is now to keep the stability it still has and the UK has turned itself now into a powder keg. This is not only about economics anymore (and even here the power scale is clearly tipped in the direction of the EU).